LMFDB - Embedded newform 3648.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3648.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.1294266574$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	3648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} -5.56155 q^{11} -3.12311 q^{13} -1.56155 q^{15} +6.68466 q^{17} -1.00000 q^{19} +1.56155 q^{21} +9.12311 q^{23} -2.56155 q^{25} -1.00000 q^{27} +8.24621 q^{29} -2.00000 q^{31} +5.56155 q^{33} -2.43845 q^{35} +8.00000 q^{37} +3.12311 q^{39} -5.12311 q^{41} +1.56155 q^{43} +1.56155 q^{45} -6.68466 q^{47} -4.56155 q^{49} -6.68466 q^{51} -4.24621 q^{53} -8.68466 q^{55} +1.00000 q^{57} -12.0000 q^{59} -6.68466 q^{61} -1.56155 q^{63} -4.87689 q^{65} -6.24621 q^{67} -9.12311 q^{69} -16.9309 q^{73} +2.56155 q^{75} +8.68466 q^{77} +11.3693 q^{79} +1.00000 q^{81} -4.00000 q^{83} +10.4384 q^{85} -8.24621 q^{87} -6.00000 q^{89} +4.87689 q^{91} +2.00000 q^{93} -1.56155 q^{95} +4.24621 q^{97} -5.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} + q^{17} - 2 q^{19} - q^{21} + 10 q^{23} - q^{25} - 2 q^{27} - 4 q^{31} + 7 q^{33} - 9 q^{35} + 16 q^{37} - 2 q^{39} - 2 q^{41} - q^{43} - q^{45} - q^{47} - 5 q^{49} - q^{51} + 8 q^{53} - 5 q^{55} + 2 q^{57} - 24 q^{59} - q^{61} + q^{63} - 18 q^{65} + 4 q^{67} - 10 q^{69} - 5 q^{73} + q^{75} + 5 q^{77} - 2 q^{79} + 2 q^{81} - 8 q^{83} + 25 q^{85} - 12 q^{89} + 18 q^{91} + 4 q^{93} + q^{95} - 8 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9 - 7 * q^11 + 2 * q^13 + q^15 + q^17 - 2 * q^19 - q^21 + 10 * q^23 - q^25 - 2 * q^27 - 4 * q^31 + 7 * q^33 - 9 * q^35 + 16 * q^37 - 2 * q^39 - 2 * q^41 - q^43 - q^45 - q^47 - 5 * q^49 - q^51 + 8 * q^53 - 5 * q^55 + 2 * q^57 - 24 * q^59 - q^61 + q^63 - 18 * q^65 + 4 * q^67 - 10 * q^69 - 5 * q^73 + q^75 + 5 * q^77 - 2 * q^79 + 2 * q^81 - 8 * q^83 + 25 * q^85 - 12 * q^89 + 18 * q^91 + 4 * q^93 + q^95 - 8 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.56155	−1.67687	−0.838436	−	0.545001i	$-0.816529\pi$
$11$	−5.56155	−1.67687	−0.838436	+	0.545001i	$0.816529\pi$
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	6.68466	1.62127	0.810634	−	0.585553i	$-0.199123\pi$
$17$	6.68466	1.62127	0.810634	+	0.585553i	$0.199123\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	9.12311	1.90230	0.951150	−	0.308731i	$-0.0999042\pi$
$23$	9.12311	1.90230	0.951150	+	0.308731i	$0.0999042\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	5.56155	0.968142
$34$	0	0
$35$	−2.43845	−0.412173
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	3.12311	0.500097
$40$	0	0
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	0	0
$43$	1.56155	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$43$	1.56155	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$44$	0	0
$45$	1.56155	0.232783
$46$	0	0
$47$	−6.68466	−0.975058	−0.487529	−	0.873107i	$-0.662102\pi$
$47$	−6.68466	−0.975058	−0.487529	+	0.873107i	$0.662102\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−6.68466	−0.936039
$52$	0	0
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	−8.68466	−1.17104
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−6.68466	−0.855883	−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466	−0.855883	−0.427941	+	0.903806i	$0.640761\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	−4.87689	−0.604904
$66$	0	0
$67$	−6.24621	−0.763096	−0.381548	−	0.924349i	$-0.624609\pi$
$67$	−6.24621	−0.763096	−0.381548	+	0.924349i	$0.624609\pi$
$68$	0	0
$69$	−9.12311	−1.09829
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−16.9309	−1.98161	−0.990804	−	0.135303i	$-0.956799\pi$
$73$	−16.9309	−1.98161	−0.990804	+	0.135303i	$0.956799\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	8.68466	0.989709
$78$	0	0
$79$	11.3693	1.27915	0.639574	−	0.768729i	$-0.279111\pi$
$79$	11.3693	1.27915	0.639574	+	0.768729i	$0.279111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	10.4384	1.13221
$86$	0	0
$87$	−8.24621	−0.884087
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.87689	0.511237
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	−1.56155	−0.160212
$96$	0	0
$97$	4.24621	0.431137	0.215569	−	0.976489i	$-0.430839\pi$
$97$	4.24621	0.431137	0.215569	+	0.976489i	$0.430839\pi$
$98$	0	0
$99$	−5.56155	−0.558957
$100$	0	0
$101$	−7.12311	−0.708776	−0.354388	−	0.935099i	$-0.615311\pi$
$101$	−7.12311	−0.708776	−0.354388	+	0.935099i	$0.615311\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	2.43845	0.237968
$106$	0	0
$107$	13.3693	1.29246	0.646230	−	0.763142i	$-0.276345\pi$
$107$	13.3693	1.29246	0.646230	+	0.763142i	$0.276345\pi$
$108$	0	0
$109$	2.24621	0.215148	0.107574	−	0.994197i	$-0.465692\pi$
$109$	2.24621	0.215148	0.107574	+	0.994197i	$0.465692\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	14.2462	1.32847
$116$	0	0
$117$	−3.12311	−0.288731
$118$	0	0
$119$	−10.4384	−0.956891
$120$	0	0
$121$	19.9309	1.81190
$122$	0	0
$123$	5.12311	0.461935
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	−1.56155	−0.137487
$130$	0	0
$131$	−20.6847	−1.80723	−0.903613	−	0.428349i	$-0.859095\pi$
$131$	−20.6847	−1.80723	−0.903613	+	0.428349i	$0.859095\pi$
$132$	0	0
$133$	1.56155	0.135404
$134$	0	0
$135$	−1.56155	−0.134397
$136$	0	0
$137$	−13.8078	−1.17968	−0.589838	−	0.807521i	$-0.700809\pi$
$137$	−13.8078	−1.17968	−0.589838	+	0.807521i	$0.700809\pi$
$138$	0	0
$139$	7.80776	0.662246	0.331123	−	0.943588i	$-0.392573\pi$
$139$	7.80776	0.662246	0.331123	+	0.943588i	$0.392573\pi$
$140$	0	0
$141$	6.68466	0.562950
$142$	0	0
$143$	17.3693	1.45250
$144$	0	0
$145$	12.8769	1.06937
$146$	0	0
$147$	4.56155	0.376231
$148$	0	0
$149$	−3.80776	−0.311944	−0.155972	−	0.987761i	$-0.549851\pi$
$149$	−3.80776	−0.311944	−0.155972	+	0.987761i	$0.549851\pi$
$150$	0	0
$151$	−17.1231	−1.39346	−0.696729	−	0.717334i	$-0.745362\pi$
$151$	−17.1231	−1.39346	−0.696729	+	0.717334i	$0.745362\pi$
$152$	0	0
$153$	6.68466	0.540423
$154$	0	0
$155$	−3.12311	−0.250854
$156$	0	0
$157$	12.2462	0.977354	0.488677	−	0.872465i	$-0.337480\pi$
$157$	12.2462	0.977354	0.488677	+	0.872465i	$0.337480\pi$
$158$	0	0
$159$	4.24621	0.336746
$160$	0	0
$161$	−14.2462	−1.12276
$162$	0	0
$163$	2.24621	0.175937	0.0879684	−	0.996123i	$-0.471963\pi$
$163$	2.24621	0.175937	0.0879684	+	0.996123i	$0.471963\pi$
$164$	0	0
$165$	8.68466	0.676100
$166$	0	0
$167$	−8.87689	−0.686915	−0.343457	−	0.939168i	$-0.611598\pi$
$167$	−8.87689	−0.686915	−0.343457	+	0.939168i	$0.611598\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	4.24621	0.322833	0.161417	−	0.986886i	$-0.448394\pi$
$173$	4.24621	0.322833	0.161417	+	0.986886i	$0.448394\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	15.1231	1.13035	0.565177	−	0.824970i	$-0.308808\pi$
$179$	15.1231	1.13035	0.565177	+	0.824970i	$0.308808\pi$
$180$	0	0
$181$	−15.1231	−1.12409	−0.562046	−	0.827106i	$-0.689986\pi$
$181$	−15.1231	−1.12409	−0.562046	+	0.827106i	$0.689986\pi$
$182$	0	0
$183$	6.68466	0.494144
$184$	0	0
$185$	12.4924	0.918461
$186$	0	0
$187$	−37.1771	−2.71866
$188$	0	0
$189$	1.56155	0.113586
$190$	0	0
$191$	−1.31534	−0.0951748	−0.0475874	−	0.998867i	$-0.515153\pi$
$191$	−1.31534	−0.0951748	−0.0475874	+	0.998867i	$0.515153\pi$
$192$	0	0
$193$	18.8769	1.35879	0.679394	−	0.733773i	$-0.262243\pi$
$193$	18.8769	1.35879	0.679394	+	0.733773i	$0.262243\pi$
$194$	0	0
$195$	4.87689	0.349242
$196$	0	0
$197$	1.36932	0.0975598	0.0487799	−	0.998810i	$-0.484467\pi$
$197$	1.36932	0.0975598	0.0487799	+	0.998810i	$0.484467\pi$
$198$	0	0
$199$	2.93087	0.207764	0.103882	−	0.994590i	$-0.466874\pi$
$199$	2.93087	0.207764	0.103882	+	0.994590i	$0.466874\pi$
$200$	0	0
$201$	6.24621	0.440574
$202$	0	0
$203$	−12.8769	−0.903781
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	9.12311	0.634100
$208$	0	0
$209$	5.56155	0.384701
$210$	0	0
$211$	−5.75379	−0.396107	−0.198054	−	0.980191i	$-0.563462\pi$
$211$	−5.75379	−0.396107	−0.198054	+	0.980191i	$0.563462\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.43845	0.166301
$216$	0	0
$217$	3.12311	0.212010
$218$	0	0
$219$	16.9309	1.14408
$220$	0	0
$221$	−20.8769	−1.40433
$222$	0	0
$223$	−11.3693	−0.761346	−0.380673	−	0.924710i	$-0.624308\pi$
$223$	−11.3693	−0.761346	−0.380673	+	0.924710i	$0.624308\pi$
$224$	0	0
$225$	−2.56155	−0.170770
$226$	0	0
$227$	8.87689	0.589180	0.294590	−	0.955624i	$-0.404817\pi$
$227$	8.87689	0.589180	0.294590	+	0.955624i	$0.404817\pi$
$228$	0	0
$229$	−3.56155	−0.235354	−0.117677	−	0.993052i	$-0.537545\pi$
$229$	−3.56155	−0.235354	−0.117677	+	0.993052i	$0.537545\pi$
$230$	0	0
$231$	−8.68466	−0.571409
$232$	0	0
$233$	−14.6847	−0.962024	−0.481012	−	0.876714i	$-0.659731\pi$
$233$	−14.6847	−0.962024	−0.481012	+	0.876714i	$0.659731\pi$
$234$	0	0
$235$	−10.4384	−0.680929
$236$	0	0
$237$	−11.3693	−0.738516
$238$	0	0
$239$	6.19224	0.400542	0.200271	−	0.979740i	$-0.435818\pi$
$239$	6.19224	0.400542	0.200271	+	0.979740i	$0.435818\pi$
$240$	0	0
$241$	−11.3693	−0.732362	−0.366181	−	0.930544i	$-0.619335\pi$
$241$	−11.3693	−0.732362	−0.366181	+	0.930544i	$0.619335\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.12311	−0.455079
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	7.80776	0.492822	0.246411	−	0.969165i	$-0.420749\pi$
$251$	7.80776	0.492822	0.246411	+	0.969165i	$0.420749\pi$
$252$	0	0
$253$	−50.7386	−3.18991
$254$	0	0
$255$	−10.4384	−0.653681
$256$	0	0
$257$	13.1231	0.818597	0.409298	−	0.912401i	$-0.365773\pi$
$257$	13.1231	0.818597	0.409298	+	0.912401i	$0.365773\pi$
$258$	0	0
$259$	−12.4924	−0.776241
$260$	0	0
$261$	8.24621	0.510428
$262$	0	0
$263$	−14.6847	−0.905495	−0.452747	−	0.891639i	$-0.649556\pi$
$263$	−14.6847	−0.905495	−0.452747	+	0.891639i	$0.649556\pi$
$264$	0	0
$265$	−6.63068	−0.407320
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−4.87689	−0.295163
$274$	0	0
$275$	14.2462	0.859079
$276$	0	0
$277$	18.6847	1.12265	0.561326	−	0.827595i	$-0.310291\pi$
$277$	18.6847	1.12265	0.561326	+	0.827595i	$0.310291\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−24.2462	−1.44641	−0.723204	−	0.690635i	$-0.757331\pi$
$281$	−24.2462	−1.44641	−0.723204	+	0.690635i	$0.757331\pi$
$282$	0	0
$283$	−28.6847	−1.70513	−0.852563	−	0.522625i	$-0.824953\pi$
$283$	−28.6847	−1.70513	−0.852563	+	0.522625i	$0.824953\pi$
$284$	0	0
$285$	1.56155	0.0924984
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	−4.24621	−0.248917
$292$	0	0
$293$	1.12311	0.0656125	0.0328063	−	0.999462i	$-0.489556\pi$
$293$	1.12311	0.0656125	0.0328063	+	0.999462i	$0.489556\pi$
$294$	0	0
$295$	−18.7386	−1.09101
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	−28.4924	−1.64776
$300$	0	0
$301$	−2.43845	−0.140550
$302$	0	0
$303$	7.12311	0.409212
$304$	0	0
$305$	−10.4384	−0.597704
$306$	0	0
$307$	4.49242	0.256396	0.128198	−	0.991749i	$-0.459081\pi$
$307$	4.49242	0.256396	0.128198	+	0.991749i	$0.459081\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	−20.9309	−1.18688	−0.593440	−	0.804878i	$-0.702231\pi$
$311$	−20.9309	−1.18688	−0.593440	+	0.804878i	$0.702231\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−2.43845	−0.137391
$316$	0	0
$317$	18.8769	1.06023	0.530116	−	0.847925i	$-0.322148\pi$
$317$	18.8769	1.06023	0.530116	+	0.847925i	$0.322148\pi$
$318$	0	0
$319$	−45.8617	−2.56776
$320$	0	0
$321$	−13.3693	−0.746203
$322$	0	0
$323$	−6.68466	−0.371944
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	−2.24621	−0.124216
$328$	0	0
$329$	10.4384	0.575490
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	−9.75379	−0.532906
$336$	0	0
$337$	20.7386	1.12971	0.564853	−	0.825192i	$-0.308933\pi$
$337$	20.7386	1.12971	0.564853	+	0.825192i	$0.308933\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	11.1231	0.602350
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−14.2462	−0.766990
$346$	0	0
$347$	−27.4233	−1.47216	−0.736080	−	0.676895i	$-0.763325\pi$
$347$	−27.4233	−1.47216	−0.736080	+	0.676895i	$0.763325\pi$
$348$	0	0
$349$	−17.3153	−0.926869	−0.463434	−	0.886131i	$-0.653383\pi$
$349$	−17.3153	−0.926869	−0.463434	+	0.886131i	$0.653383\pi$
$350$	0	0
$351$	3.12311	0.166699
$352$	0	0
$353$	−28.2462	−1.50339	−0.751697	−	0.659509i	$-0.770764\pi$
$353$	−28.2462	−1.50339	−0.751697	+	0.659509i	$0.770764\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	10.4384	0.552461
$358$	0	0
$359$	0.438447	0.0231404	0.0115702	−	0.999933i	$-0.496317\pi$
$359$	0.438447	0.0231404	0.0115702	+	0.999933i	$0.496317\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−19.9309	−1.04610
$364$	0	0
$365$	−26.4384	−1.38385
$366$	0	0
$367$	2.24621	0.117251	0.0586256	−	0.998280i	$-0.481328\pi$
$367$	2.24621	0.117251	0.0586256	+	0.998280i	$0.481328\pi$
$368$	0	0
$369$	−5.12311	−0.266698
$370$	0	0
$371$	6.63068	0.344248
$372$	0	0
$373$	36.4924	1.88951	0.944753	−	0.327783i	$-0.106302\pi$
$373$	36.4924	1.88951	0.944753	+	0.327783i	$0.106302\pi$
$374$	0	0
$375$	11.8078	0.609750
$376$	0	0
$377$	−25.7538	−1.32639
$378$	0	0
$379$	20.8769	1.07237	0.536187	−	0.844099i	$-0.319864\pi$
$379$	20.8769	1.07237	0.536187	+	0.844099i	$0.319864\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	12.8769	0.657979	0.328989	−	0.944334i	$-0.393292\pi$
$383$	12.8769	0.657979	0.328989	+	0.944334i	$0.393292\pi$
$384$	0	0
$385$	13.5616	0.691161
$386$	0	0
$387$	1.56155	0.0793782
$388$	0	0
$389$	−18.9309	−0.959833	−0.479917	−	0.877314i	$-0.659333\pi$
$389$	−18.9309	−0.959833	−0.479917	+	0.877314i	$0.659333\pi$
$390$	0	0
$391$	60.9848	3.08414
$392$	0	0
$393$	20.6847	1.04340
$394$	0	0
$395$	17.7538	0.893290
$396$	0	0
$397$	22.6847	1.13851	0.569255	−	0.822161i	$-0.307232\pi$
$397$	22.6847	1.13851	0.569255	+	0.822161i	$0.307232\pi$
$398$	0	0
$399$	−1.56155	−0.0781754
$400$	0	0
$401$	−0.630683	−0.0314948	−0.0157474	−	0.999876i	$-0.505013\pi$
$401$	−0.630683	−0.0314948	−0.0157474	+	0.999876i	$0.505013\pi$
$402$	0	0
$403$	6.24621	0.311146
$404$	0	0
$405$	1.56155	0.0775942
$406$	0	0
$407$	−44.4924	−2.20541
$408$	0	0
$409$	−0.246211	−0.0121744	−0.00608718	−	0.999981i	$-0.501938\pi$
$409$	−0.246211	−0.0121744	−0.00608718	+	0.999981i	$0.501938\pi$
$410$	0	0
$411$	13.8078	0.681087
$412$	0	0
$413$	18.7386	0.922068
$414$	0	0
$415$	−6.24621	−0.306614
$416$	0	0
$417$	−7.80776	−0.382348
$418$	0	0
$419$	26.7386	1.30627	0.653134	−	0.757242i	$-0.273454\pi$
$419$	26.7386	1.30627	0.653134	+	0.757242i	$0.273454\pi$
$420$	0	0
$421$	−4.87689	−0.237685	−0.118843	−	0.992913i	$-0.537918\pi$
$421$	−4.87689	−0.237685	−0.118843	+	0.992913i	$0.537918\pi$
$422$	0	0
$423$	−6.68466	−0.325019
$424$	0	0
$425$	−17.1231	−0.830593
$426$	0	0
$427$	10.4384	0.505152
$428$	0	0
$429$	−17.3693	−0.838599
$430$	0	0
$431$	−11.6155	−0.559500	−0.279750	−	0.960073i	$-0.590252\pi$
$431$	−11.6155	−0.559500	−0.279750	+	0.960073i	$0.590252\pi$
$432$	0	0
$433$	−8.24621	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$433$	−8.24621	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$434$	0	0
$435$	−12.8769	−0.617400
$436$	0	0
$437$	−9.12311	−0.436417
$438$	0	0
$439$	28.2462	1.34812	0.674059	−	0.738677i	$-0.264549\pi$
$439$	28.2462	1.34812	0.674059	+	0.738677i	$0.264549\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	−12.1922	−0.579271	−0.289635	−	0.957137i	$-0.593534\pi$
$443$	−12.1922	−0.579271	−0.289635	+	0.957137i	$0.593534\pi$
$444$	0	0
$445$	−9.36932	−0.444148
$446$	0	0
$447$	3.80776	0.180101
$448$	0	0
$449$	35.3693	1.66918	0.834591	−	0.550871i	$-0.185704\pi$
$449$	35.3693	1.66918	0.834591	+	0.550871i	$0.185704\pi$
$450$	0	0
$451$	28.4924	1.34166
$452$	0	0
$453$	17.1231	0.804514
$454$	0	0
$455$	7.61553	0.357021
$456$	0	0
$457$	−40.0540	−1.87365	−0.936823	−	0.349804i	$-0.886248\pi$
$457$	−40.0540	−1.87365	−0.936823	+	0.349804i	$0.886248\pi$
$458$	0	0
$459$	−6.68466	−0.312013
$460$	0	0
$461$	−32.3002	−1.50437	−0.752185	−	0.658952i	$-0.771000\pi$
$461$	−32.3002	−1.50437	−0.752185	+	0.658952i	$0.771000\pi$
$462$	0	0
$463$	31.8078	1.47823	0.739116	−	0.673578i	$-0.235243\pi$
$463$	31.8078	1.47823	0.739116	+	0.673578i	$0.235243\pi$
$464$	0	0
$465$	3.12311	0.144831
$466$	0	0
$467$	−10.9309	−0.505820	−0.252910	−	0.967490i	$-0.581388\pi$
$467$	−10.9309	−0.505820	−0.252910	+	0.967490i	$0.581388\pi$
$468$	0	0
$469$	9.75379	0.450388
$470$	0	0
$471$	−12.2462	−0.564276
$472$	0	0
$473$	−8.68466	−0.399321
$474$	0	0
$475$	2.56155	0.117532
$476$	0	0
$477$	−4.24621	−0.194421
$478$	0	0
$479$	15.3693	0.702242	0.351121	−	0.936330i	$-0.385801\pi$
$479$	15.3693	0.702242	0.351121	+	0.936330i	$0.385801\pi$
$480$	0	0
$481$	−24.9848	−1.13921
$482$	0	0
$483$	14.2462	0.648225
$484$	0	0
$485$	6.63068	0.301084
$486$	0	0
$487$	−33.1231	−1.50095	−0.750476	−	0.660898i	$-0.770176\pi$
$487$	−33.1231	−1.50095	−0.750476	+	0.660898i	$0.770176\pi$
$488$	0	0
$489$	−2.24621	−0.101577
$490$	0	0
$491$	14.7386	0.665145	0.332573	−	0.943078i	$-0.392083\pi$
$491$	14.7386	0.665145	0.332573	+	0.943078i	$0.392083\pi$
$492$	0	0
$493$	55.1231	2.48262
$494$	0	0
$495$	−8.68466	−0.390346
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.17708	0.410823	0.205411	−	0.978676i	$-0.434147\pi$
$499$	9.17708	0.410823	0.205411	+	0.978676i	$0.434147\pi$
$500$	0	0
$501$	8.87689	0.396590
$502$	0	0
$503$	25.6155	1.14214	0.571070	−	0.820901i	$-0.306528\pi$
$503$	25.6155	1.14214	0.571070	+	0.820901i	$0.306528\pi$
$504$	0	0
$505$	−11.1231	−0.494972
$506$	0	0
$507$	3.24621	0.144169
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	26.4384	1.16957
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−9.36932	−0.412861
$516$	0	0
$517$	37.1771	1.63505
$518$	0	0
$519$	−4.24621	−0.186388
$520$	0	0
$521$	9.61553	0.421264	0.210632	−	0.977565i	$-0.432448\pi$
$521$	9.61553	0.421264	0.210632	+	0.977565i	$0.432448\pi$
$522$	0	0
$523$	−17.3693	−0.759507	−0.379754	−	0.925088i	$-0.623991\pi$
$523$	−17.3693	−0.759507	−0.379754	+	0.925088i	$0.623991\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−13.3693	−0.582377
$528$	0	0
$529$	60.2311	2.61874
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	20.8769	0.902587
$536$	0	0
$537$	−15.1231	−0.652610
$538$	0	0
$539$	25.3693	1.09273
$540$	0	0
$541$	17.8078	0.765616	0.382808	−	0.923828i	$-0.374957\pi$
$541$	17.8078	0.765616	0.382808	+	0.923828i	$0.374957\pi$
$542$	0	0
$543$	15.1231	0.648995
$544$	0	0
$545$	3.50758	0.150248
$546$	0	0
$547$	0.384472	0.0164388	0.00821942	−	0.999966i	$-0.497384\pi$
$547$	0.384472	0.0164388	0.00821942	+	0.999966i	$0.497384\pi$
$548$	0	0
$549$	−6.68466	−0.285294
$550$	0	0
$551$	−8.24621	−0.351300
$552$	0	0
$553$	−17.7538	−0.754968
$554$	0	0
$555$	−12.4924	−0.530274
$556$	0	0
$557$	−45.5616	−1.93050	−0.965252	−	0.261319i	$-0.915843\pi$
$557$	−45.5616	−1.93050	−0.965252	+	0.261319i	$0.915843\pi$
$558$	0	0
$559$	−4.87689	−0.206271
$560$	0	0
$561$	37.1771	1.56962
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−3.12311	−0.131390
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	−18.4924	−0.775243	−0.387621	−	0.921819i	$-0.626703\pi$
$569$	−18.4924	−0.775243	−0.387621	+	0.921819i	$0.626703\pi$
$570$	0	0
$571$	−6.73863	−0.282003	−0.141002	−	0.990009i	$-0.545032\pi$
$571$	−6.73863	−0.282003	−0.141002	+	0.990009i	$0.545032\pi$
$572$	0	0
$573$	1.31534	0.0549492
$574$	0	0
$575$	−23.3693	−0.974568
$576$	0	0
$577$	−11.5616	−0.481314	−0.240657	−	0.970610i	$-0.577363\pi$
$577$	−11.5616	−0.481314	−0.240657	+	0.970610i	$0.577363\pi$
$578$	0	0
$579$	−18.8769	−0.784497
$580$	0	0
$581$	6.24621	0.259137
$582$	0	0
$583$	23.6155	0.978055
$584$	0	0
$585$	−4.87689	−0.201635
$586$	0	0
$587$	18.0540	0.745167	0.372584	−	0.927999i	$-0.378472\pi$
$587$	18.0540	0.745167	0.372584	+	0.927999i	$0.378472\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	−1.36932	−0.0563262
$592$	0	0
$593$	38.4924	1.58069	0.790347	−	0.612659i	$-0.209900\pi$
$593$	38.4924	1.58069	0.790347	+	0.612659i	$0.209900\pi$
$594$	0	0
$595$	−16.3002	−0.668242
$596$	0	0
$597$	−2.93087	−0.119953
$598$	0	0
$599$	15.5076	0.633622	0.316811	−	0.948489i	$-0.397388\pi$
$599$	15.5076	0.633622	0.316811	+	0.948489i	$0.397388\pi$
$600$	0	0
$601$	39.8617	1.62599	0.812997	−	0.582268i	$-0.197834\pi$
$601$	39.8617	1.62599	0.812997	+	0.582268i	$0.197834\pi$
$602$	0	0
$603$	−6.24621	−0.254365
$604$	0	0
$605$	31.1231	1.26533
$606$	0	0
$607$	−22.9848	−0.932926	−0.466463	−	0.884541i	$-0.654472\pi$
$607$	−22.9848	−0.932926	−0.466463	+	0.884541i	$0.654472\pi$
$608$	0	0
$609$	12.8769	0.521798
$610$	0	0
$611$	20.8769	0.844589
$612$	0	0
$613$	−38.7926	−1.56682	−0.783409	−	0.621506i	$-0.786521\pi$
$613$	−38.7926	−1.56682	−0.783409	+	0.621506i	$0.786521\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	−14.3002	−0.575704	−0.287852	−	0.957675i	$-0.592941\pi$
$617$	−14.3002	−0.575704	−0.287852	+	0.957675i	$0.592941\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−9.12311	−0.366098
$622$	0	0
$623$	9.36932	0.375374
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	−5.56155	−0.222107
$628$	0	0
$629$	53.4773	2.13228
$630$	0	0
$631$	22.9309	0.912864	0.456432	−	0.889758i	$-0.349127\pi$
$631$	22.9309	0.912864	0.456432	+	0.889758i	$0.349127\pi$
$632$	0	0
$633$	5.75379	0.228693
$634$	0	0
$635$	−28.1080	−1.11543
$636$	0	0
$637$	14.2462	0.564455
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.7386	−1.60908	−0.804540	−	0.593899i	$-0.797588\pi$
$641$	−40.7386	−1.60908	−0.804540	+	0.593899i	$0.797588\pi$
$642$	0	0
$643$	−9.94602	−0.392233	−0.196116	−	0.980581i	$-0.562833\pi$
$643$	−9.94602	−0.392233	−0.196116	+	0.980581i	$0.562833\pi$
$644$	0	0
$645$	−2.43845	−0.0960138
$646$	0	0
$647$	−21.8078	−0.857352	−0.428676	−	0.903458i	$-0.641020\pi$
$647$	−21.8078	−0.857352	−0.428676	+	0.903458i	$0.641020\pi$
$648$	0	0
$649$	66.7386	2.61972
$650$	0	0
$651$	−3.12311	−0.122404
$652$	0	0
$653$	−40.3002	−1.57707	−0.788534	−	0.614991i	$-0.789160\pi$
$653$	−40.3002	−1.57707	−0.788534	+	0.614991i	$0.789160\pi$
$654$	0	0
$655$	−32.3002	−1.26207
$656$	0	0
$657$	−16.9309	−0.660536
$658$	0	0
$659$	−21.3693	−0.832430	−0.416215	−	0.909266i	$-0.636644\pi$
$659$	−21.3693	−0.832430	−0.416215	+	0.909266i	$0.636644\pi$
$660$	0	0
$661$	42.7386	1.66234	0.831170	−	0.556018i	$-0.187672\pi$
$661$	42.7386	1.66234	0.831170	+	0.556018i	$0.187672\pi$
$662$	0	0
$663$	20.8769	0.810791
$664$	0	0
$665$	2.43845	0.0945589
$666$	0	0
$667$	75.2311	2.91296
$668$	0	0
$669$	11.3693	0.439563
$670$	0	0
$671$	37.1771	1.43521
$672$	0	0
$673$	−31.8617	−1.22818	−0.614090	−	0.789236i	$-0.710477\pi$
$673$	−31.8617	−1.22818	−0.614090	+	0.789236i	$0.710477\pi$
$674$	0	0
$675$	2.56155	0.0985942
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	−6.63068	−0.254462
$680$	0	0
$681$	−8.87689	−0.340163
$682$	0	0
$683$	9.86174	0.377349	0.188674	−	0.982040i	$-0.439581\pi$
$683$	9.86174	0.377349	0.188674	+	0.982040i	$0.439581\pi$
$684$	0	0
$685$	−21.5616	−0.823825
$686$	0	0
$687$	3.56155	0.135882
$688$	0	0
$689$	13.2614	0.505218
$690$	0	0
$691$	−21.0691	−0.801507	−0.400754	−	0.916186i	$-0.631252\pi$
$691$	−21.0691	−0.801507	−0.400754	+	0.916186i	$0.631252\pi$
$692$	0	0
$693$	8.68466	0.329903
$694$	0	0
$695$	12.1922	0.462478
$696$	0	0
$697$	−34.2462	−1.29717
$698$	0	0
$699$	14.6847	0.555425
$700$	0	0
$701$	−23.1231	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$701$	−23.1231	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	10.4384	0.393135
$706$	0	0
$707$	11.1231	0.418327
$708$	0	0
$709$	−9.50758	−0.357065	−0.178532	−	0.983934i	$-0.557135\pi$
$709$	−9.50758	−0.357065	−0.178532	+	0.983934i	$0.557135\pi$
$710$	0	0
$711$	11.3693	0.426383
$712$	0	0
$713$	−18.2462	−0.683326
$714$	0	0
$715$	27.1231	1.01435
$716$	0	0
$717$	−6.19224	−0.231253
$718$	0	0
$719$	37.4233	1.39565	0.697827	−	0.716267i	$-0.254151\pi$
$719$	37.4233	1.39565	0.697827	+	0.716267i	$0.254151\pi$
$720$	0	0
$721$	9.36932	0.348932
$722$	0	0
$723$	11.3693	0.422829
$724$	0	0
$725$	−21.1231	−0.784492
$726$	0	0
$727$	13.5616	0.502970	0.251485	−	0.967861i	$-0.419081\pi$
$727$	13.5616	0.502970	0.251485	+	0.967861i	$0.419081\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.4384	0.386080
$732$	0	0
$733$	18.4924	0.683033	0.341517	−	0.939876i	$-0.389059\pi$
$733$	18.4924	0.683033	0.341517	+	0.939876i	$0.389059\pi$
$734$	0	0
$735$	7.12311	0.262740
$736$	0	0
$737$	34.7386	1.27961
$738$	0	0
$739$	−25.1771	−0.926154	−0.463077	−	0.886318i	$-0.653255\pi$
$739$	−25.1771	−0.926154	−0.463077	+	0.886318i	$0.653255\pi$
$740$	0	0
$741$	−3.12311	−0.114730
$742$	0	0
$743$	40.4924	1.48552	0.742761	−	0.669556i	$-0.233516\pi$
$743$	40.4924	1.48552	0.742761	+	0.669556i	$0.233516\pi$
$744$	0	0
$745$	−5.94602	−0.217845
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−20.8769	−0.762825
$750$	0	0
$751$	33.6155	1.22665	0.613324	−	0.789831i	$-0.289832\pi$
$751$	33.6155	1.22665	0.613324	+	0.789831i	$0.289832\pi$
$752$	0	0
$753$	−7.80776	−0.284531
$754$	0	0
$755$	−26.7386	−0.973119
$756$	0	0
$757$	32.0540	1.16502	0.582511	−	0.812823i	$-0.302070\pi$
$757$	32.0540	1.16502	0.582511	+	0.812823i	$0.302070\pi$
$758$	0	0
$759$	50.7386	1.84170
$760$	0	0
$761$	47.6695	1.72802	0.864009	−	0.503476i	$-0.167946\pi$
$761$	47.6695	1.72802	0.864009	+	0.503476i	$0.167946\pi$
$762$	0	0
$763$	−3.50758	−0.126983
$764$	0	0
$765$	10.4384	0.377403
$766$	0	0
$767$	37.4773	1.35323
$768$	0	0
$769$	−34.3002	−1.23690	−0.618448	−	0.785826i	$-0.712238\pi$
$769$	−34.3002	−1.23690	−0.618448	+	0.785826i	$0.712238\pi$
$770$	0	0
$771$	−13.1231	−0.472617
$772$	0	0
$773$	−17.6155	−0.633587	−0.316793	−	0.948495i	$-0.602606\pi$
$773$	−17.6155	−0.633587	−0.316793	+	0.948495i	$0.602606\pi$
$774$	0	0
$775$	5.12311	0.184027
$776$	0	0
$777$	12.4924	0.448163
$778$	0	0
$779$	5.12311	0.183554
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.24621	−0.294696
$784$	0	0
$785$	19.1231	0.682533
$786$	0	0
$787$	−10.2462	−0.365238	−0.182619	−	0.983184i	$-0.558457\pi$
$787$	−10.2462	−0.365238	−0.182619	+	0.983184i	$0.558457\pi$
$788$	0	0
$789$	14.6847	0.522788
$790$	0	0
$791$	3.12311	0.111045
$792$	0	0
$793$	20.8769	0.741360
$794$	0	0
$795$	6.63068	0.235166
$796$	0	0
$797$	−41.6155	−1.47410	−0.737049	−	0.675840i	$-0.763781\pi$
$797$	−41.6155	−1.47410	−0.737049	+	0.675840i	$0.763781\pi$
$798$	0	0
$799$	−44.6847	−1.58083
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	94.1619	3.32290
$804$	0	0
$805$	−22.2462	−0.784076
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−7.94602	−0.279367	−0.139684	−	0.990196i	$-0.544609\pi$
$809$	−7.94602	−0.279367	−0.139684	+	0.990196i	$0.544609\pi$
$810$	0	0
$811$	35.1231	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$811$	35.1231	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	3.50758	0.122865
$816$	0	0
$817$	−1.56155	−0.0546318
$818$	0	0
$819$	4.87689	0.170412
$820$	0	0
$821$	1.94602	0.0679167	0.0339584	−	0.999423i	$-0.489189\pi$
$821$	1.94602	0.0679167	0.0339584	+	0.999423i	$0.489189\pi$
$822$	0	0
$823$	−40.3002	−1.40478	−0.702388	−	0.711794i	$-0.747883\pi$
$823$	−40.3002	−1.40478	−0.702388	+	0.711794i	$0.747883\pi$
$824$	0	0
$825$	−14.2462	−0.495989
$826$	0	0
$827$	14.7386	0.512513	0.256256	−	0.966609i	$-0.417511\pi$
$827$	14.7386	0.512513	0.256256	+	0.966609i	$0.417511\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	−18.6847	−0.648164
$832$	0	0
$833$	−30.4924	−1.05650
$834$	0	0
$835$	−13.8617	−0.479705
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	−49.8617	−1.72142	−0.860709	−	0.509097i	$-0.829979\pi$
$839$	−49.8617	−1.72142	−0.860709	+	0.509097i	$0.829979\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	24.2462	0.835084
$844$	0	0
$845$	−5.06913	−0.174383
$846$	0	0
$847$	−31.1231	−1.06940
$848$	0	0
$849$	28.6847	0.984455
$850$	0	0
$851$	72.9848	2.50189
$852$	0	0
$853$	7.26137	0.248624	0.124312	−	0.992243i	$-0.460328\pi$
$853$	7.26137	0.248624	0.124312	+	0.992243i	$0.460328\pi$
$854$	0	0
$855$	−1.56155	−0.0534040
$856$	0	0
$857$	15.8617	0.541827	0.270913	−	0.962604i	$-0.412674\pi$
$857$	15.8617	0.541827	0.270913	+	0.962604i	$0.412674\pi$
$858$	0	0
$859$	28.3002	0.965590	0.482795	−	0.875733i	$-0.339622\pi$
$859$	28.3002	0.965590	0.482795	+	0.875733i	$0.339622\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	6.63068	0.225450
$866$	0	0
$867$	−27.6847	−0.940220
$868$	0	0
$869$	−63.2311	−2.14497
$870$	0	0
$871$	19.5076	0.660989
$872$	0	0
$873$	4.24621	0.143712
$874$	0	0
$875$	18.4384	0.623333
$876$	0	0
$877$	22.6307	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$877$	22.6307	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$878$	0	0
$879$	−1.12311	−0.0378814
$880$	0	0
$881$	−30.3002	−1.02084	−0.510420	−	0.859925i	$-0.670510\pi$
$881$	−30.3002	−1.02084	−0.510420	+	0.859925i	$0.670510\pi$
$882$	0	0
$883$	35.3153	1.18846	0.594228	−	0.804297i	$-0.297458\pi$
$883$	35.3153	1.18846	0.594228	+	0.804297i	$0.297458\pi$
$884$	0	0
$885$	18.7386	0.629892
$886$	0	0
$887$	51.2311	1.72017	0.860085	−	0.510150i	$-0.170410\pi$
$887$	51.2311	1.72017	0.860085	+	0.510150i	$0.170410\pi$
$888$	0	0
$889$	28.1080	0.942710
$890$	0	0
$891$	−5.56155	−0.186319
$892$	0	0
$893$	6.68466	0.223694
$894$	0	0
$895$	23.6155	0.789380
$896$	0	0
$897$	28.4924	0.951334
$898$	0	0
$899$	−16.4924	−0.550053
$900$	0	0
$901$	−28.3845	−0.945624
$902$	0	0
$903$	2.43845	0.0811464
$904$	0	0
$905$	−23.6155	−0.785007
$906$	0	0
$907$	43.1231	1.43188	0.715940	−	0.698162i	$-0.245999\pi$
$907$	43.1231	1.43188	0.715940	+	0.698162i	$0.245999\pi$
$908$	0	0
$909$	−7.12311	−0.236259
$910$	0	0
$911$	−7.12311	−0.235999	−0.118000	−	0.993014i	$-0.537648\pi$
$911$	−7.12311	−0.235999	−0.118000	+	0.993014i	$0.537648\pi$
$912$	0	0
$913$	22.2462	0.736242
$914$	0	0
$915$	10.4384	0.345084
$916$	0	0
$917$	32.3002	1.06665
$918$	0	0
$919$	11.5076	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$919$	11.5076	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$920$	0	0
$921$	−4.49242	−0.148030
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.4924	−0.673787
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	4.56155	0.149499
$932$	0	0
$933$	20.9309	0.685246
$934$	0	0
$935$	−58.0540	−1.89857
$936$	0	0
$937$	−12.4384	−0.406346	−0.203173	−	0.979143i	$-0.565125\pi$
$937$	−12.4384	−0.406346	−0.203173	+	0.979143i	$0.565125\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−46.7386	−1.52202
$944$	0	0
$945$	2.43845	0.0793227
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	52.8769	1.71646
$950$	0	0
$951$	−18.8769	−0.612125
$952$	0	0
$953$	29.6155	0.959341	0.479671	−	0.877449i	$-0.340756\pi$
$953$	29.6155	0.959341	0.479671	+	0.877449i	$0.340756\pi$
$954$	0	0
$955$	−2.05398	−0.0664651
$956$	0	0
$957$	45.8617	1.48250
$958$	0	0
$959$	21.5616	0.696259
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	13.3693	0.430820
$964$	0	0
$965$	29.4773	0.948907
$966$	0	0
$967$	22.7386	0.731225	0.365613	−	0.930767i	$-0.380860\pi$
$967$	22.7386	0.731225	0.365613	+	0.930767i	$0.380860\pi$
$968$	0	0
$969$	6.68466	0.214742
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−12.1922	−0.390865
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	−58.9848	−1.88709	−0.943546	−	0.331241i	$-0.892533\pi$
$977$	−58.9848	−1.88709	−0.943546	+	0.331241i	$0.892533\pi$
$978$	0	0
$979$	33.3693	1.06649
$980$	0	0
$981$	2.24621	0.0717160
$982$	0	0
$983$	−18.7386	−0.597670	−0.298835	−	0.954305i	$-0.596598\pi$
$983$	−18.7386	−0.597670	−0.298835	+	0.954305i	$0.596598\pi$
$984$	0	0
$985$	2.13826	0.0681306
$986$	0	0
$987$	−10.4384	−0.332259
$988$	0	0
$989$	14.2462	0.453003
$990$	0	0
$991$	35.8617	1.13919	0.569593	−	0.821927i	$-0.307101\pi$
$991$	35.8617	1.13919	0.569593	+	0.821927i	$0.307101\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.57671	0.145091
$996$	0	0
$997$	−30.7926	−0.975212	−0.487606	−	0.873064i	$-0.662130\pi$
$997$	−30.7926	−0.975212	−0.487606	+	0.873064i	$0.662130\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3648.2.a.bl.1.2		2
4.3	odd	2		3648.2.a.br.1.2		2
8.3	odd	2		912.2.a.m.1.1		2
8.5	even	2		456.2.a.f.1.1	✓	2
24.5	odd	2		1368.2.a.k.1.2		2
24.11	even	2		2736.2.a.z.1.2		2
152.37	odd	2		8664.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.f.1.1	✓	2	8.5	even	2
912.2.a.m.1.1		2	8.3	odd	2
1368.2.a.k.1.2		2	24.5	odd	2
2736.2.a.z.1.2		2	24.11	even	2
3648.2.a.bl.1.2		2	1.1	even	1	trivial
3648.2.a.br.1.2		2	4.3	odd	2
8664.2.a.r.1.1		2	152.37	odd	2

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:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3648.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} -5.56155 q^{11} -3.12311 q^{13} -1.56155 q^{15} +6.68466 q^{17} -1.00000 q^{19} +1.56155 q^{21} +9.12311 q^{23} -2.56155 q^{25} -1.00000 q^{27} +8.24621 q^{29} -2.00000 q^{31} +5.56155 q^{33} -2.43845 q^{35} +8.00000 q^{37} +3.12311 q^{39} -5.12311 q^{41} +1.56155 q^{43} +1.56155 q^{45} -6.68466 q^{47} -4.56155 q^{49} -6.68466 q^{51} -4.24621 q^{53} -8.68466 q^{55} +1.00000 q^{57} -12.0000 q^{59} -6.68466 q^{61} -1.56155 q^{63} -4.87689 q^{65} -6.24621 q^{67} -9.12311 q^{69} -16.9309 q^{73} +2.56155 q^{75} +8.68466 q^{77} +11.3693 q^{79} +1.00000 q^{81} -4.00000 q^{83} +10.4384 q^{85} -8.24621 q^{87} -6.00000 q^{89} +4.87689 q^{91} +2.00000 q^{93} -1.56155 q^{95} +4.24621 q^{97} -5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} + q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} + q^{17} - 2 q^{19} - q^{21} + 10 q^{23} - q^{25} - 2 q^{27} - 4 q^{31} + 7 q^{33} - 9 q^{35} + 16 q^{37} - 2 q^{39} - 2 q^{41} - q^{43} - q^{45} - q^{47} - 5 q^{49} - q^{51} + 8 q^{53} - 5 q^{55} + 2 q^{57} - 24 q^{59} - q^{61} + q^{63} - 18 q^{65} + 4 q^{67} - 10 q^{69} - 5 q^{73} + q^{75} + 5 q^{77} - 2 q^{79} + 2 q^{81} - 8 q^{83} + 25 q^{85} - 12 q^{89} + 18 q^{91} + 4 q^{93} + q^{95} - 8 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + q^7 + 2 * q^9 - 7 * q^11 + 2 * q^13 + q^15 + q^17 - 2 * q^19 - q^21 + 10 * q^23 - q^25 - 2 * q^27 - 4 * q^31 + 7 * q^33 - 9 * q^35 + 16 * q^37 - 2 * q^39 - 2 * q^41 - q^43 - q^45 - q^47 - 5 * q^49 - q^51 + 8 * q^53 - 5 * q^55 + 2 * q^57 - 24 * q^59 - q^61 + q^63 - 18 * q^65 + 4 * q^67 - 10 * q^69 - 5 * q^73 + q^75 + 5 * q^77 - 2 * q^79 + 2 * q^81 - 8 * q^83 + 25 * q^85 - 12 * q^89 + 18 * q^91 + 4 * q^93 + q^95 - 8 * q^97 - 7 * q^99

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