LMFDB - Embedded newform 572.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(1,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("572.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.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:	$4.56744299562$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	572.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302776 q^{3} -3.30278 q^{7} -2.90833 q^{9} +O(q^{10})$	$q+0.302776 q^{3} -3.30278 q^{7} -2.90833 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.60555 q^{17} -1.69722 q^{19} -1.00000 q^{21} -5.30278 q^{23} -5.00000 q^{25} -1.78890 q^{27} +9.21110 q^{29} -2.60555 q^{31} -0.302776 q^{33} +5.21110 q^{37} +0.302776 q^{39} +5.30278 q^{41} +0.605551 q^{43} -6.00000 q^{47} +3.90833 q^{49} -1.39445 q^{51} -8.30278 q^{53} -0.513878 q^{57} +7.39445 q^{59} -11.8167 q^{61} +9.60555 q^{63} -8.60555 q^{67} -1.60555 q^{69} -10.6056 q^{71} +7.51388 q^{73} -1.51388 q^{75} +3.30278 q^{77} -0.788897 q^{79} +8.18335 q^{81} +8.51388 q^{83} +2.78890 q^{87} -3.30278 q^{91} -0.788897 q^{93} +17.2111 q^{97} +2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 7 q^{19} - 2 q^{21} - 7 q^{23} - 10 q^{25} - 18 q^{27} + 4 q^{29} + 2 q^{31} + 3 q^{33} - 4 q^{37} - 3 q^{39} + 7 q^{41} - 6 q^{43} - 12 q^{47} - 3 q^{49} - 10 q^{51} - 13 q^{53} + 17 q^{57} + 22 q^{59} - 2 q^{61} + 12 q^{63} - 10 q^{67} + 4 q^{69} - 14 q^{71} - 3 q^{73} + 15 q^{75} + 3 q^{77} - 16 q^{79} + 38 q^{81} - q^{83} + 20 q^{87} - 3 q^{91} - 16 q^{93} + 20 q^{97} - 5 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 7 * q^19 - 2 * q^21 - 7 * q^23 - 10 * q^25 - 18 * q^27 + 4 * q^29 + 2 * q^31 + 3 * q^33 - 4 * q^37 - 3 * q^39 + 7 * q^41 - 6 * q^43 - 12 * q^47 - 3 * q^49 - 10 * q^51 - 13 * q^53 + 17 * q^57 + 22 * q^59 - 2 * q^61 + 12 * q^63 - 10 * q^67 + 4 * q^69 - 14 * q^71 - 3 * q^73 + 15 * q^75 + 3 * q^77 - 16 * q^79 + 38 * q^81 - q^83 + 20 * q^87 - 3 * q^91 - 16 * q^93 + 20 * q^97 - 5 * 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
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−3.30278	−1.24833	−0.624166	−	0.781292i	$-0.714561\pi$
$7$	−3.30278	−1.24833	−0.624166	+	0.781292i	$0.714561\pi$
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	0	0
$19$	−1.69722	−0.389370	−0.194685	−	0.980866i	$-0.562368\pi$
$19$	−1.69722	−0.389370	−0.194685	+	0.980866i	$0.562368\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−5.30278	−1.10571	−0.552853	−	0.833279i	$-0.686461\pi$
$23$	−5.30278	−1.10571	−0.552853	+	0.833279i	$0.686461\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−1.78890	−0.344273
$28$	0	0
$29$	9.21110	1.71046	0.855229	−	0.518250i	$-0.173416\pi$
$29$	9.21110	1.71046	0.855229	+	0.518250i	$0.173416\pi$
$30$	0	0
$31$	−2.60555	−0.467971	−0.233985	−	0.972240i	$-0.575177\pi$
$31$	−2.60555	−0.467971	−0.233985	+	0.972240i	$0.575177\pi$
$32$	0	0
$33$	−0.302776	−0.0527065
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.21110	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$37$	5.21110	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$38$	0	0
$39$	0.302776	0.0484829
$40$	0	0
$41$	5.30278	0.828154	0.414077	−	0.910242i	$-0.364104\pi$
$41$	5.30278	0.828154	0.414077	+	0.910242i	$0.364104\pi$
$42$	0	0
$43$	0.605551	0.0923457	0.0461729	−	0.998933i	$-0.485297\pi$
$43$	0.605551	0.0923457	0.0461729	+	0.998933i	$0.485297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	3.90833	0.558332
$50$	0	0
$51$	−1.39445	−0.195262
$52$	0	0
$53$	−8.30278	−1.14047	−0.570237	−	0.821480i	$-0.693149\pi$
$53$	−8.30278	−1.14047	−0.570237	+	0.821480i	$0.693149\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.513878	−0.0680648
$58$	0	0
$59$	7.39445	0.962675	0.481338	−	0.876535i	$-0.340151\pi$
$59$	7.39445	0.962675	0.481338	+	0.876535i	$0.340151\pi$
$60$	0	0
$61$	−11.8167	−1.51297	−0.756484	−	0.654012i	$-0.773084\pi$
$61$	−11.8167	−1.51297	−0.756484	+	0.654012i	$0.773084\pi$
$62$	0	0
$63$	9.60555	1.21019
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.60555	−1.05134	−0.525668	−	0.850690i	$-0.676184\pi$
$67$	−8.60555	−1.05134	−0.525668	+	0.850690i	$0.676184\pi$
$68$	0	0
$69$	−1.60555	−0.193286
$70$	0	0
$71$	−10.6056	−1.25865	−0.629324	−	0.777143i	$-0.716668\pi$
$71$	−10.6056	−1.25865	−0.629324	+	0.777143i	$0.716668\pi$
$72$	0	0
$73$	7.51388	0.879433	0.439716	−	0.898137i	$-0.355079\pi$
$73$	7.51388	0.879433	0.439716	+	0.898137i	$0.355079\pi$
$74$	0	0
$75$	−1.51388	−0.174808
$76$	0	0
$77$	3.30278	0.376386
$78$	0	0
$79$	−0.788897	−0.0887579	−0.0443789	−	0.999015i	$-0.514131\pi$
$79$	−0.788897	−0.0887579	−0.0443789	+	0.999015i	$0.514131\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	8.51388	0.934520	0.467260	−	0.884120i	$-0.345241\pi$
$83$	8.51388	0.934520	0.467260	+	0.884120i	$0.345241\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.78890	0.299001
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.30278	−0.346225
$92$	0	0
$93$	−0.788897	−0.0818049
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.2111	1.74752	0.873761	−	0.486355i	$-0.161674\pi$
$97$	17.2111	1.74752	0.873761	+	0.486355i	$0.161674\pi$
$98$	0	0
$99$	2.90833	0.292298
$100$	0	0
$101$	−7.81665	−0.777786	−0.388893	−	0.921283i	$-0.627142\pi$
$101$	−7.81665	−0.777786	−0.388893	+	0.921283i	$0.627142\pi$
$102$	0	0
$103$	−7.90833	−0.779231	−0.389615	−	0.920978i	$-0.627392\pi$
$103$	−7.90833	−0.779231	−0.389615	+	0.920978i	$0.627392\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	14.9083	1.42796	0.713979	−	0.700167i	$-0.246891\pi$
$109$	14.9083	1.42796	0.713979	+	0.700167i	$0.246891\pi$
$110$	0	0
$111$	1.57779	0.149758
$112$	0	0
$113$	12.9083	1.21431	0.607157	−	0.794582i	$-0.292310\pi$
$113$	12.9083	1.21431	0.607157	+	0.794582i	$0.292310\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.90833	−0.268875
$118$	0	0
$119$	15.2111	1.39440
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.60555	0.144768
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	0.183346	0.0161427
$130$	0	0
$131$	9.21110	0.804778	0.402389	−	0.915469i	$-0.368180\pi$
$131$	9.21110	0.804778	0.402389	+	0.915469i	$0.368180\pi$
$132$	0	0
$133$	5.60555	0.486063
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−11.3944	−0.966465	−0.483232	−	0.875492i	$-0.660537\pi$
$139$	−11.3944	−0.966465	−0.483232	+	0.875492i	$0.660537\pi$
$140$	0	0
$141$	−1.81665	−0.152990
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.18335	0.0976007
$148$	0	0
$149$	22.3305	1.82939	0.914694	−	0.404147i	$-0.132431\pi$
$149$	22.3305	1.82939	0.914694	+	0.404147i	$0.132431\pi$
$150$	0	0
$151$	−19.6333	−1.59774	−0.798868	−	0.601506i	$-0.794567\pi$
$151$	−19.6333	−1.59774	−0.798868	+	0.601506i	$0.794567\pi$
$152$	0	0
$153$	13.3944	1.08288
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.9083	−1.11001	−0.555003	−	0.831849i	$-0.687283\pi$
$157$	−13.9083	−1.11001	−0.555003	+	0.831849i	$0.687283\pi$
$158$	0	0
$159$	−2.51388	−0.199364
$160$	0	0
$161$	17.5139	1.38029
$162$	0	0
$163$	−11.3944	−0.892482	−0.446241	−	0.894913i	$-0.647238\pi$
$163$	−11.3944	−0.892482	−0.446241	+	0.894913i	$0.647238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.09167	0.394005	0.197003	−	0.980403i	$-0.436879\pi$
$167$	5.09167	0.394005	0.197003	+	0.980403i	$0.436879\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.93608	0.377472
$172$	0	0
$173$	3.21110	0.244136	0.122068	−	0.992522i	$-0.461047\pi$
$173$	3.21110	0.244136	0.122068	+	0.992522i	$0.461047\pi$
$174$	0	0
$175$	16.5139	1.24833
$176$	0	0
$177$	2.23886	0.168283
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−12.3028	−0.914458	−0.457229	−	0.889349i	$-0.651158\pi$
$181$	−12.3028	−0.914458	−0.457229	+	0.889349i	$0.651158\pi$
$182$	0	0
$183$	−3.57779	−0.264478
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.60555	0.336791
$188$	0	0
$189$	5.90833	0.429768
$190$	0	0
$191$	17.5139	1.26726	0.633630	−	0.773636i	$-0.281564\pi$
$191$	17.5139	1.26726	0.633630	+	0.773636i	$0.281564\pi$
$192$	0	0
$193$	−21.3028	−1.53341	−0.766704	−	0.642001i	$-0.778104\pi$
$193$	−21.3028	−1.53341	−0.766704	+	0.642001i	$0.778104\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.09167	−0.362767	−0.181383	−	0.983412i	$-0.558057\pi$
$197$	−5.09167	−0.362767	−0.181383	+	0.983412i	$0.558057\pi$
$198$	0	0
$199$	−27.9361	−1.98034	−0.990168	−	0.139882i	$-0.955328\pi$
$199$	−27.9361	−1.98034	−0.990168	+	0.139882i	$0.955328\pi$
$200$	0	0
$201$	−2.60555	−0.183781
$202$	0	0
$203$	−30.4222	−2.13522
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.4222	1.07192
$208$	0	0
$209$	1.69722	0.117399
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	−3.21110	−0.220021
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.60555	0.584183
$218$	0	0
$219$	2.27502	0.153732
$220$	0	0
$221$	−4.60555	−0.309803
$222$	0	0
$223$	−23.8167	−1.59488	−0.797441	−	0.603398i	$-0.793813\pi$
$223$	−23.8167	−1.59488	−0.797441	+	0.603398i	$0.793813\pi$
$224$	0	0
$225$	14.5416	0.969442
$226$	0	0
$227$	−4.11943	−0.273416	−0.136708	−	0.990611i	$-0.543652\pi$
$227$	−4.11943	−0.273416	−0.136708	+	0.990611i	$0.543652\pi$
$228$	0	0
$229$	−0.788897	−0.0521318	−0.0260659	−	0.999660i	$-0.508298\pi$
$229$	−0.788897	−0.0521318	−0.0260659	+	0.999660i	$0.508298\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−10.6056	−0.694793	−0.347396	−	0.937718i	$-0.612934\pi$
$233$	−10.6056	−0.694793	−0.347396	+	0.937718i	$0.612934\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.238859	−0.0155156
$238$	0	0
$239$	12.4861	0.807660	0.403830	−	0.914834i	$-0.367679\pi$
$239$	12.4861	0.807660	0.403830	+	0.914834i	$0.367679\pi$
$240$	0	0
$241$	30.3305	1.95376	0.976881	−	0.213785i	$-0.0685793\pi$
$241$	30.3305	1.95376	0.976881	+	0.213785i	$0.0685793\pi$
$242$	0	0
$243$	7.84441	0.503219
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.69722	−0.107992
$248$	0	0
$249$	2.57779	0.163361
$250$	0	0
$251$	−23.7250	−1.49751	−0.748754	−	0.662848i	$-0.769347\pi$
$251$	−23.7250	−1.49751	−0.748754	+	0.662848i	$0.769347\pi$
$252$	0	0
$253$	5.30278	0.333383
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.11943	0.444098	0.222049	−	0.975036i	$-0.428726\pi$
$257$	7.11943	0.444098	0.222049	+	0.975036i	$0.428726\pi$
$258$	0	0
$259$	−17.2111	−1.06945
$260$	0	0
$261$	−26.7889	−1.65819
$262$	0	0
$263$	13.3944	0.825937	0.412969	−	0.910745i	$-0.364492\pi$
$263$	13.3944	0.825937	0.412969	+	0.910745i	$0.364492\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.7250	−1.08071	−0.540356	−	0.841437i	$-0.681710\pi$
$269$	−17.7250	−1.08071	−0.540356	+	0.841437i	$0.681710\pi$
$270$	0	0
$271$	−1.69722	−0.103099	−0.0515495	−	0.998670i	$-0.516416\pi$
$271$	−1.69722	−0.103099	−0.0515495	+	0.998670i	$0.516416\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−31.2111	−1.87529	−0.937647	−	0.347590i	$-0.887000\pi$
$277$	−31.2111	−1.87529	−0.937647	+	0.347590i	$0.887000\pi$
$278$	0	0
$279$	7.57779	0.453671
$280$	0	0
$281$	−13.3305	−0.795233	−0.397616	−	0.917552i	$-0.630163\pi$
$281$	−13.3305	−0.795233	−0.397616	+	0.917552i	$0.630163\pi$
$282$	0	0
$283$	11.2111	0.666431	0.333215	−	0.942851i	$-0.391866\pi$
$283$	11.2111	0.666431	0.333215	+	0.942851i	$0.391866\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.5139	−1.03381
$288$	0	0
$289$	4.21110	0.247712
$290$	0	0
$291$	5.21110	0.305480
$292$	0	0
$293$	−21.6333	−1.26383	−0.631916	−	0.775037i	$-0.717731\pi$
$293$	−21.6333	−1.26383	−0.631916	+	0.775037i	$0.717731\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.78890	0.103802
$298$	0	0
$299$	−5.30278	−0.306667
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	−2.36669	−0.135963
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.4222	−0.594827	−0.297413	−	0.954749i	$-0.596124\pi$
$307$	−10.4222	−0.594827	−0.297413	+	0.954749i	$0.596124\pi$
$308$	0	0
$309$	−2.39445	−0.136215
$310$	0	0
$311$	7.33053	0.415676	0.207838	−	0.978163i	$-0.433357\pi$
$311$	7.33053	0.415676	0.207838	+	0.978163i	$0.433357\pi$
$312$	0	0
$313$	−21.7250	−1.22797	−0.613984	−	0.789318i	$-0.710434\pi$
$313$	−21.7250	−1.22797	−0.613984	+	0.789318i	$0.710434\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.39445	0.0783200	0.0391600	−	0.999233i	$-0.487532\pi$
$317$	1.39445	0.0783200	0.0391600	+	0.999233i	$0.487532\pi$
$318$	0	0
$319$	−9.21110	−0.515723
$320$	0	0
$321$	1.81665	0.101396
$322$	0	0
$323$	7.81665	0.434930
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	4.51388	0.249618
$328$	0	0
$329$	19.8167	1.09253
$330$	0	0
$331$	3.39445	0.186576	0.0932879	−	0.995639i	$-0.470262\pi$
$331$	3.39445	0.186576	0.0932879	+	0.995639i	$0.470262\pi$
$332$	0	0
$333$	−15.1556	−0.830521
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.81665	−0.316853	−0.158427	−	0.987371i	$-0.550642\pi$
$337$	−5.81665	−0.316853	−0.158427	+	0.987371i	$0.550642\pi$
$338$	0	0
$339$	3.90833	0.212271
$340$	0	0
$341$	2.60555	0.141099
$342$	0	0
$343$	10.2111	0.551348
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.4500	−1.90305	−0.951527	−	0.307566i	$-0.900486\pi$
$347$	−35.4500	−1.90305	−0.951527	+	0.307566i	$0.900486\pi$
$348$	0	0
$349$	10.5139	0.562795	0.281397	−	0.959591i	$-0.409202\pi$
$349$	10.5139	0.562795	0.281397	+	0.959591i	$0.409202\pi$
$350$	0	0
$351$	−1.78890	−0.0954843
$352$	0	0
$353$	−20.2389	−1.07721	−0.538603	−	0.842560i	$-0.681048\pi$
$353$	−20.2389	−1.07721	−0.538603	+	0.842560i	$0.681048\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.60555	0.243752
$358$	0	0
$359$	−0.697224	−0.0367981	−0.0183990	−	0.999831i	$-0.505857\pi$
$359$	−0.697224	−0.0367981	−0.0183990	+	0.999831i	$0.505857\pi$
$360$	0	0
$361$	−16.1194	−0.848391
$362$	0	0
$363$	0.302776	0.0158916
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.5416	1.28106	0.640531	−	0.767932i	$-0.278714\pi$
$367$	24.5416	1.28106	0.640531	+	0.767932i	$0.278714\pi$
$368$	0	0
$369$	−15.4222	−0.802848
$370$	0	0
$371$	27.4222	1.42369
$372$	0	0
$373$	−22.8444	−1.18284	−0.591419	−	0.806364i	$-0.701432\pi$
$373$	−22.8444	−1.18284	−0.591419	+	0.806364i	$0.701432\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.21110	0.474396
$378$	0	0
$379$	−11.8167	−0.606981	−0.303490	−	0.952835i	$-0.598152\pi$
$379$	−11.8167	−0.606981	−0.303490	+	0.952835i	$0.598152\pi$
$380$	0	0
$381$	−3.02776	−0.155117
$382$	0	0
$383$	21.6333	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$383$	21.6333	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.76114	−0.0895238
$388$	0	0
$389$	14.7250	0.746586	0.373293	−	0.927713i	$-0.378229\pi$
$389$	14.7250	0.746586	0.373293	+	0.927713i	$0.378229\pi$
$390$	0	0
$391$	24.4222	1.23508
$392$	0	0
$393$	2.78890	0.140681
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.0278	−1.05535	−0.527676	−	0.849445i	$-0.676937\pi$
$397$	−21.0278	−1.05535	−0.527676	+	0.849445i	$0.676937\pi$
$398$	0	0
$399$	1.69722	0.0849675
$400$	0	0
$401$	−4.18335	−0.208906	−0.104453	−	0.994530i	$-0.533309\pi$
$401$	−4.18335	−0.208906	−0.104453	+	0.994530i	$0.533309\pi$
$402$	0	0
$403$	−2.60555	−0.129792
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.21110	−0.258305
$408$	0	0
$409$	16.7889	0.830158	0.415079	−	0.909785i	$-0.363754\pi$
$409$	16.7889	0.830158	0.415079	+	0.909785i	$0.363754\pi$
$410$	0	0
$411$	−1.81665	−0.0896089
$412$	0	0
$413$	−24.4222	−1.20174
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.44996	−0.168945
$418$	0	0
$419$	28.7527	1.40466	0.702332	−	0.711850i	$-0.252142\pi$
$419$	28.7527	1.40466	0.702332	+	0.711850i	$0.252142\pi$
$420$	0	0
$421$	7.02776	0.342512	0.171256	−	0.985227i	$-0.445217\pi$
$421$	7.02776	0.342512	0.171256	+	0.985227i	$0.445217\pi$
$422$	0	0
$423$	17.4500	0.848446
$424$	0	0
$425$	23.0278	1.11701
$426$	0	0
$427$	39.0278	1.88869
$428$	0	0
$429$	−0.302776	−0.0146181
$430$	0	0
$431$	−5.72498	−0.275763	−0.137881	−	0.990449i	$-0.544029\pi$
$431$	−5.72498	−0.275763	−0.137881	+	0.990449i	$0.544029\pi$
$432$	0	0
$433$	6.11943	0.294081	0.147041	−	0.989130i	$-0.453025\pi$
$433$	6.11943	0.294081	0.147041	+	0.989130i	$0.453025\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.00000	0.430528
$438$	0	0
$439$	−12.7889	−0.610381	−0.305190	−	0.952291i	$-0.598720\pi$
$439$	−12.7889	−0.610381	−0.305190	+	0.952291i	$0.598720\pi$
$440$	0	0
$441$	−11.3667	−0.541271
$442$	0	0
$443$	−39.3583	−1.86997	−0.934984	−	0.354689i	$-0.884587\pi$
$443$	−39.3583	−1.86997	−0.934984	+	0.354689i	$0.884587\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.76114	0.319791
$448$	0	0
$449$	−28.6056	−1.34998	−0.674990	−	0.737827i	$-0.735852\pi$
$449$	−28.6056	−1.34998	−0.674990	+	0.737827i	$0.735852\pi$
$450$	0	0
$451$	−5.30278	−0.249698
$452$	0	0
$453$	−5.94449	−0.279296
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.8806	0.742862	0.371431	−	0.928461i	$-0.378867\pi$
$457$	15.8806	0.742862	0.371431	+	0.928461i	$0.378867\pi$
$458$	0	0
$459$	8.23886	0.384557
$460$	0	0
$461$	2.30278	0.107251	0.0536255	−	0.998561i	$-0.482922\pi$
$461$	2.30278	0.107251	0.0536255	+	0.998561i	$0.482922\pi$
$462$	0	0
$463$	16.2389	0.754684	0.377342	−	0.926074i	$-0.376838\pi$
$463$	16.2389	0.754684	0.377342	+	0.926074i	$0.376838\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.63331	−0.168129	−0.0840647	−	0.996460i	$-0.526790\pi$
$467$	−3.63331	−0.168129	−0.0840647	+	0.996460i	$0.526790\pi$
$468$	0	0
$469$	28.4222	1.31242
$470$	0	0
$471$	−4.21110	−0.194037
$472$	0	0
$473$	−0.605551	−0.0278433
$474$	0	0
$475$	8.48612	0.389370
$476$	0	0
$477$	24.1472	1.10562
$478$	0	0
$479$	42.4222	1.93832	0.969160	−	0.246432i	$-0.0792583\pi$
$479$	42.4222	1.93832	0.969160	+	0.246432i	$0.0792583\pi$
$480$	0	0
$481$	5.21110	0.237606
$482$	0	0
$483$	5.30278	0.241285
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−33.0278	−1.49663	−0.748315	−	0.663343i	$-0.769137\pi$
$487$	−33.0278	−1.49663	−0.748315	+	0.663343i	$0.769137\pi$
$488$	0	0
$489$	−3.44996	−0.156013
$490$	0	0
$491$	8.23886	0.371814	0.185907	−	0.982567i	$-0.440478\pi$
$491$	8.23886	0.371814	0.185907	+	0.982567i	$0.440478\pi$
$492$	0	0
$493$	−42.4222	−1.91060
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.0278	1.57121
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	1.54163	0.0688752
$502$	0	0
$503$	−1.81665	−0.0810006	−0.0405003	−	0.999180i	$-0.512895\pi$
$503$	−1.81665	−0.0810006	−0.0405003	+	0.999180i	$0.512895\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.302776	0.0134467
$508$	0	0
$509$	−1.81665	−0.0805218	−0.0402609	−	0.999189i	$-0.512819\pi$
$509$	−1.81665	−0.0805218	−0.0402609	+	0.999189i	$0.512819\pi$
$510$	0	0
$511$	−24.8167	−1.09782
$512$	0	0
$513$	3.03616	0.134050
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0.972244	0.0426768
$520$	0	0
$521$	−33.9083	−1.48555	−0.742775	−	0.669541i	$-0.766491\pi$
$521$	−33.9083	−1.48555	−0.742775	+	0.669541i	$0.766491\pi$
$522$	0	0
$523$	−5.39445	−0.235883	−0.117941	−	0.993021i	$-0.537629\pi$
$523$	−5.39445	−0.235883	−0.117941	+	0.993021i	$0.537629\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	5.11943	0.222584
$530$	0	0
$531$	−21.5055	−0.933258
$532$	0	0
$533$	5.30278	0.229689
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.63331	0.156789
$538$	0	0
$539$	−3.90833	−0.168344
$540$	0	0
$541$	−17.1194	−0.736022	−0.368011	−	0.929821i	$-0.619961\pi$
$541$	−17.1194	−0.736022	−0.368011	+	0.929821i	$0.619961\pi$
$542$	0	0
$543$	−3.72498	−0.159854
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.81665	−0.248702	−0.124351	−	0.992238i	$-0.539685\pi$
$547$	−5.81665	−0.248702	−0.124351	+	0.992238i	$0.539685\pi$
$548$	0	0
$549$	34.3667	1.46673
$550$	0	0
$551$	−15.6333	−0.666001
$552$	0	0
$553$	2.60555	0.110799
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.3305	0.437719	0.218859	−	0.975756i	$-0.429766\pi$
$557$	10.3305	0.437719	0.218859	+	0.975756i	$0.429766\pi$
$558$	0	0
$559$	0.605551	0.0256121
$560$	0	0
$561$	1.39445	0.0588737
$562$	0	0
$563$	43.8167	1.84665	0.923326	−	0.384017i	$-0.125460\pi$
$563$	43.8167	1.84665	0.923326	+	0.384017i	$0.125460\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−27.0278	−1.13506
$568$	0	0
$569$	28.6056	1.19921	0.599604	−	0.800297i	$-0.295325\pi$
$569$	28.6056	1.19921	0.599604	+	0.800297i	$0.295325\pi$
$570$	0	0
$571$	9.81665	0.410814	0.205407	−	0.978677i	$-0.434148\pi$
$571$	9.81665	0.410814	0.205407	+	0.978677i	$0.434148\pi$
$572$	0	0
$573$	5.30278	0.221527
$574$	0	0
$575$	26.5139	1.10571
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	−6.44996	−0.268051
$580$	0	0
$581$	−28.1194	−1.16659
$582$	0	0
$583$	8.30278	0.343866
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.18335	0.172665	0.0863326	−	0.996266i	$-0.472485\pi$
$587$	4.18335	0.172665	0.0863326	+	0.996266i	$0.472485\pi$
$588$	0	0
$589$	4.42221	0.182214
$590$	0	0
$591$	−1.54163	−0.0634144
$592$	0	0
$593$	1.11943	0.0459695	0.0229847	−	0.999736i	$-0.492683\pi$
$593$	1.11943	0.0459695	0.0229847	+	0.999736i	$0.492683\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.45837	−0.346178
$598$	0	0
$599$	−46.1194	−1.88439	−0.942194	−	0.335067i	$-0.891241\pi$
$599$	−46.1194	−1.88439	−0.942194	+	0.335067i	$0.891241\pi$
$600$	0	0
$601$	32.4222	1.32253	0.661265	−	0.750153i	$-0.270020\pi$
$601$	32.4222	1.32253	0.661265	+	0.750153i	$0.270020\pi$
$602$	0	0
$603$	25.0278	1.01921
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.18335	−0.332152	−0.166076	−	0.986113i	$-0.553110\pi$
$607$	−8.18335	−0.332152	−0.166076	+	0.986113i	$0.553110\pi$
$608$	0	0
$609$	−9.21110	−0.373253
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−5.33053	−0.215298	−0.107649	−	0.994189i	$-0.534332\pi$
$613$	−5.33053	−0.215298	−0.107649	+	0.994189i	$0.534332\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.8444	−1.00020	−0.500099	−	0.865968i	$-0.666703\pi$
$617$	−24.8444	−1.00020	−0.500099	+	0.865968i	$0.666703\pi$
$618$	0	0
$619$	38.8444	1.56129	0.780644	−	0.624976i	$-0.214891\pi$
$619$	38.8444	1.56129	0.780644	+	0.624976i	$0.214891\pi$
$620$	0	0
$621$	9.48612	0.380665
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0.513878	0.0205223
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	36.0555	1.43535	0.717674	−	0.696380i	$-0.245207\pi$
$631$	36.0555	1.43535	0.717674	+	0.696380i	$0.245207\pi$
$632$	0	0
$633$	0.605551	0.0240685
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.90833	0.154854
$638$	0	0
$639$	30.8444	1.22019
$640$	0	0
$641$	−9.90833	−0.391355	−0.195678	−	0.980668i	$-0.562691\pi$
$641$	−9.90833	−0.391355	−0.195678	+	0.980668i	$0.562691\pi$
$642$	0	0
$643$	−11.8167	−0.466003	−0.233002	−	0.972476i	$-0.574855\pi$
$643$	−11.8167	−0.466003	−0.233002	+	0.972476i	$0.574855\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.9638	1.37457	0.687285	−	0.726388i	$-0.258802\pi$
$647$	34.9638	1.37457	0.687285	+	0.726388i	$0.258802\pi$
$648$	0	0
$649$	−7.39445	−0.290258
$650$	0	0
$651$	2.60555	0.102120
$652$	0	0
$653$	−15.2111	−0.595256	−0.297628	−	0.954682i	$-0.596196\pi$
$653$	−15.2111	−0.595256	−0.297628	+	0.954682i	$0.596196\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−21.8528	−0.852559
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−22.8444	−0.888545	−0.444272	−	0.895892i	$-0.646538\pi$
$661$	−22.8444	−0.888545	−0.444272	+	0.895892i	$0.646538\pi$
$662$	0	0
$663$	−1.39445	−0.0541559
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−48.8444	−1.89126
$668$	0	0
$669$	−7.21110	−0.278797
$670$	0	0
$671$	11.8167	0.456177
$672$	0	0
$673$	−14.6056	−0.563003	−0.281501	−	0.959561i	$-0.590832\pi$
$673$	−14.6056	−0.563003	−0.281501	+	0.959561i	$0.590832\pi$
$674$	0	0
$675$	8.94449	0.344273
$676$	0	0
$677$	43.2666	1.66287	0.831436	−	0.555621i	$-0.187520\pi$
$677$	43.2666	1.66287	0.831436	+	0.555621i	$0.187520\pi$
$678$	0	0
$679$	−56.8444	−2.18149
$680$	0	0
$681$	−1.24726	−0.0477952
$682$	0	0
$683$	27.6333	1.05736	0.528680	−	0.848821i	$-0.322687\pi$
$683$	27.6333	1.05736	0.528680	+	0.848821i	$0.322687\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.238859	−0.00911304
$688$	0	0
$689$	−8.30278	−0.316311
$690$	0	0
$691$	33.8167	1.28645	0.643223	−	0.765679i	$-0.277597\pi$
$691$	33.8167	1.28645	0.643223	+	0.765679i	$0.277597\pi$
$692$	0	0
$693$	−9.60555	−0.364885
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.4222	−0.925057
$698$	0	0
$699$	−3.21110	−0.121455
$700$	0	0
$701$	−23.0278	−0.869746	−0.434873	−	0.900492i	$-0.643207\pi$
$701$	−23.0278	−0.869746	−0.434873	+	0.900492i	$0.643207\pi$
$702$	0	0
$703$	−8.84441	−0.333573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.8167	0.970935
$708$	0	0
$709$	−2.60555	−0.0978535	−0.0489268	−	0.998802i	$-0.515580\pi$
$709$	−2.60555	−0.0978535	−0.0489268	+	0.998802i	$0.515580\pi$
$710$	0	0
$711$	2.29437	0.0860457
$712$	0	0
$713$	13.8167	0.517438
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.78049	0.141185
$718$	0	0
$719$	−5.57779	−0.208017	−0.104008	−	0.994576i	$-0.533167\pi$
$719$	−5.57779	−0.208017	−0.104008	+	0.994576i	$0.533167\pi$
$720$	0	0
$721$	26.1194	0.972738
$722$	0	0
$723$	9.18335	0.341532
$724$	0	0
$725$	−46.0555	−1.71046
$726$	0	0
$727$	34.5139	1.28005	0.640024	−	0.768355i	$-0.278924\pi$
$727$	34.5139	1.28005	0.640024	+	0.768355i	$0.278924\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	−2.78890	−0.103151
$732$	0	0
$733$	−41.1194	−1.51878	−0.759390	−	0.650635i	$-0.774503\pi$
$733$	−41.1194	−1.51878	−0.759390	+	0.650635i	$0.774503\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.60555	0.316990
$738$	0	0
$739$	−7.48612	−0.275381	−0.137691	−	0.990475i	$-0.543968\pi$
$739$	−7.48612	−0.275381	−0.137691	+	0.990475i	$0.543968\pi$
$740$	0	0
$741$	−0.513878	−0.0188778
$742$	0	0
$743$	12.8444	0.471216	0.235608	−	0.971848i	$-0.424292\pi$
$743$	12.8444	0.471216	0.235608	+	0.971848i	$0.424292\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−24.7611	−0.905963
$748$	0	0
$749$	−19.8167	−0.724085
$750$	0	0
$751$	−28.9083	−1.05488	−0.527440	−	0.849592i	$-0.676848\pi$
$751$	−28.9083	−1.05488	−0.527440	+	0.849592i	$0.676848\pi$
$752$	0	0
$753$	−7.18335	−0.261776
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.0917	−0.657553	−0.328777	−	0.944408i	$-0.606636\pi$
$757$	−18.0917	−0.657553	−0.328777	+	0.944408i	$0.606636\pi$
$758$	0	0
$759$	1.60555	0.0582778
$760$	0	0
$761$	0.275019	0.00996944	0.00498472	−	0.999988i	$-0.498413\pi$
$761$	0.275019	0.00996944	0.00498472	+	0.999988i	$0.498413\pi$
$762$	0	0
$763$	−49.2389	−1.78257
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.39445	0.266998
$768$	0	0
$769$	11.9083	0.429425	0.214713	−	0.976677i	$-0.431119\pi$
$769$	11.9083	0.429425	0.214713	+	0.976677i	$0.431119\pi$
$770$	0	0
$771$	2.15559	0.0776317
$772$	0	0
$773$	8.23886	0.296331	0.148166	−	0.988963i	$-0.452663\pi$
$773$	8.23886	0.296331	0.148166	+	0.988963i	$0.452663\pi$
$774$	0	0
$775$	13.0278	0.467971
$776$	0	0
$777$	−5.21110	−0.186947
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	10.6056	0.379496
$782$	0	0
$783$	−16.4777	−0.588866
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−37.9083	−1.35129	−0.675643	−	0.737229i	$-0.736134\pi$
$787$	−37.9083	−1.35129	−0.675643	+	0.737229i	$0.736134\pi$
$788$	0	0
$789$	4.05551	0.144380
$790$	0	0
$791$	−42.6333	−1.51587
$792$	0	0
$793$	−11.8167	−0.419622
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	27.6333	0.977596
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.51388	−0.265159
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.36669	−0.188917
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	19.3028	0.677812	0.338906	−	0.940820i	$-0.389943\pi$
$811$	19.3028	0.677812	0.338906	+	0.940820i	$0.389943\pi$
$812$	0	0
$813$	−0.513878	−0.0180225
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.02776	−0.0359566
$818$	0	0
$819$	9.60555	0.335645
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−30.5139	−1.06365	−0.531823	−	0.846855i	$-0.678493\pi$
$823$	−30.5139	−1.06365	−0.531823	+	0.846855i	$0.678493\pi$
$824$	0	0
$825$	1.51388	0.0527065
$826$	0	0
$827$	−17.9361	−0.623699	−0.311849	−	0.950132i	$-0.600948\pi$
$827$	−17.9361	−0.623699	−0.311849	+	0.950132i	$0.600948\pi$
$828$	0	0
$829$	25.7250	0.893466	0.446733	−	0.894667i	$-0.352587\pi$
$829$	25.7250	0.893466	0.446733	+	0.894667i	$0.352587\pi$
$830$	0	0
$831$	−9.44996	−0.327816
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.66106	0.161110
$838$	0	0
$839$	36.4222	1.25743	0.628717	−	0.777634i	$-0.283580\pi$
$839$	36.4222	1.25743	0.628717	+	0.777634i	$0.283580\pi$
$840$	0	0
$841$	55.8444	1.92567
$842$	0	0
$843$	−4.03616	−0.139013
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.30278	−0.113485
$848$	0	0
$849$	3.39445	0.116497
$850$	0	0
$851$	−27.6333	−0.947258
$852$	0	0
$853$	13.0917	0.448250	0.224125	−	0.974560i	$-0.428048\pi$
$853$	13.0917	0.448250	0.224125	+	0.974560i	$0.428048\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.9722	−0.443123	−0.221562	−	0.975146i	$-0.571115\pi$
$857$	−12.9722	−0.443123	−0.221562	+	0.975146i	$0.571115\pi$
$858$	0	0
$859$	37.5139	1.27996	0.639979	−	0.768393i	$-0.278943\pi$
$859$	37.5139	1.27996	0.639979	+	0.768393i	$0.278943\pi$
$860$	0	0
$861$	−5.30278	−0.180718
$862$	0	0
$863$	−40.6056	−1.38223	−0.691115	−	0.722745i	$-0.742880\pi$
$863$	−40.6056	−1.38223	−0.691115	+	0.722745i	$0.742880\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.27502	0.0433019
$868$	0	0
$869$	0.788897	0.0267615
$870$	0	0
$871$	−8.60555	−0.291588
$872$	0	0
$873$	−50.0555	−1.69412
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.1194	0.814455	0.407228	−	0.913327i	$-0.366496\pi$
$877$	24.1194	0.814455	0.407228	+	0.913327i	$0.366496\pi$
$878$	0	0
$879$	−6.55004	−0.220927
$880$	0	0
$881$	−25.8806	−0.871939	−0.435969	−	0.899962i	$-0.643594\pi$
$881$	−25.8806	−0.871939	−0.435969	+	0.899962i	$0.643594\pi$
$882$	0	0
$883$	31.9361	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$883$	31.9361	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.2111	−1.51804	−0.759020	−	0.651067i	$-0.774322\pi$
$887$	−45.2111	−1.51804	−0.759020	+	0.651067i	$0.774322\pi$
$888$	0	0
$889$	33.0278	1.10772
$890$	0	0
$891$	−8.18335	−0.274152
$892$	0	0
$893$	10.1833	0.340773
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.60555	−0.0536078
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	38.2389	1.27392
$902$	0	0
$903$	−0.605551	−0.0201515
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.11943	0.302806	0.151403	−	0.988472i	$-0.451621\pi$
$907$	9.11943	0.302806	0.151403	+	0.988472i	$0.451621\pi$
$908$	0	0
$909$	22.7334	0.754019
$910$	0	0
$911$	−5.30278	−0.175689	−0.0878444	−	0.996134i	$-0.527998\pi$
$911$	−5.30278	−0.175689	−0.0878444	+	0.996134i	$0.527998\pi$
$912$	0	0
$913$	−8.51388	−0.281768
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−30.4222	−1.00463
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	−3.15559	−0.103980
$922$	0	0
$923$	−10.6056	−0.349086
$924$	0	0
$925$	−26.0555	−0.856700
$926$	0	0
$927$	23.0000	0.755419
$928$	0	0
$929$	26.2389	0.860869	0.430435	−	0.902622i	$-0.358360\pi$
$929$	26.2389	0.860869	0.430435	+	0.902622i	$0.358360\pi$
$930$	0	0
$931$	−6.63331	−0.217398
$932$	0	0
$933$	2.21951	0.0726634
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	−6.57779	−0.214658
$940$	0	0
$941$	−22.5416	−0.734836	−0.367418	−	0.930056i	$-0.619758\pi$
$941$	−22.5416	−0.734836	−0.367418	+	0.930056i	$0.619758\pi$
$942$	0	0
$943$	−28.1194	−0.915695
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.2111	−1.07922	−0.539608	−	0.841916i	$-0.681428\pi$
$947$	−33.2111	−1.07922	−0.539608	+	0.841916i	$0.681428\pi$
$948$	0	0
$949$	7.51388	0.243911
$950$	0	0
$951$	0.422205	0.0136909
$952$	0	0
$953$	15.6333	0.506413	0.253206	−	0.967412i	$-0.418515\pi$
$953$	15.6333	0.506413	0.253206	+	0.967412i	$0.418515\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.78890	−0.0901523
$958$	0	0
$959$	19.8167	0.639913
$960$	0	0
$961$	−24.2111	−0.781003
$962$	0	0
$963$	−17.4500	−0.562317
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53.3305	−1.71499	−0.857497	−	0.514489i	$-0.827982\pi$
$967$	−53.3305	−1.71499	−0.857497	+	0.514489i	$0.827982\pi$
$968$	0	0
$969$	2.36669	0.0760291
$970$	0	0
$971$	6.48612	0.208150	0.104075	−	0.994569i	$-0.466812\pi$
$971$	6.48612	0.208150	0.104075	+	0.994569i	$0.466812\pi$
$972$	0	0
$973$	37.6333	1.20647
$974$	0	0
$975$	−1.51388	−0.0484829
$976$	0	0
$977$	−5.57779	−0.178449	−0.0892247	−	0.996012i	$-0.528439\pi$
$977$	−5.57779	−0.178449	−0.0892247	+	0.996012i	$0.528439\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−43.3583	−1.38432
$982$	0	0
$983$	12.4222	0.396207	0.198103	−	0.980181i	$-0.436522\pi$
$983$	12.4222	0.396207	0.198103	+	0.980181i	$0.436522\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	−3.21110	−0.102107
$990$	0	0
$991$	31.0917	0.987660	0.493830	−	0.869559i	$-0.335597\pi$
$991$	31.0917	0.987660	0.493830	+	0.869559i	$0.335597\pi$
$992$	0	0
$993$	1.02776	0.0326149
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.39445	0.297525	0.148763	−	0.988873i	$-0.452471\pi$
$997$	9.39445	0.297525	0.148763	+	0.988873i	$0.452471\pi$
$998$	0	0
$999$	−9.32213	−0.294939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.a.b.1.2	✓	2
3.2	odd	2		5148.2.a.h.1.1		2
4.3	odd	2		2288.2.a.r.1.1		2
8.3	odd	2		9152.2.a.bg.1.2		2
8.5	even	2		9152.2.a.bq.1.1		2
11.10	odd	2		6292.2.a.k.1.2		2
13.12	even	2		7436.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.b.1.2	✓	2	1.1	even	1	trivial
2288.2.a.r.1.1		2	4.3	odd	2
5148.2.a.h.1.1		2	3.2	odd	2
6292.2.a.k.1.2		2	11.10	odd	2
7436.2.a.e.1.2		2	13.12	even	2
9152.2.a.bg.1.2		2	8.3	odd	2
9152.2.a.bq.1.1		2	8.5	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.a (trivial)

\(f(q)\)	\(=\)	\(q+0.302776 q^{3} -3.30278 q^{7} -2.90833 q^{9} +O(q^{10})\)	\(q+0.302776 q^{3} -3.30278 q^{7} -2.90833 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.60555 q^{17} -1.69722 q^{19} -1.00000 q^{21} -5.30278 q^{23} -5.00000 q^{25} -1.78890 q^{27} +9.21110 q^{29} -2.60555 q^{31} -0.302776 q^{33} +5.21110 q^{37} +0.302776 q^{39} +5.30278 q^{41} +0.605551 q^{43} -6.00000 q^{47} +3.90833 q^{49} -1.39445 q^{51} -8.30278 q^{53} -0.513878 q^{57} +7.39445 q^{59} -11.8167 q^{61} +9.60555 q^{63} -8.60555 q^{67} -1.60555 q^{69} -10.6056 q^{71} +7.51388 q^{73} -1.51388 q^{75} +3.30278 q^{77} -0.788897 q^{79} +8.18335 q^{81} +8.51388 q^{83} +2.78890 q^{87} -3.30278 q^{91} -0.788897 q^{93} +17.2111 q^{97} +2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} - 3 q^{7} + 5 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 7 q^{19} - 2 q^{21} - 7 q^{23} - 10 q^{25} - 18 q^{27} + 4 q^{29} + 2 q^{31} + 3 q^{33} - 4 q^{37} - 3 q^{39} + 7 q^{41} - 6 q^{43} - 12 q^{47} - 3 q^{49} - 10 q^{51} - 13 q^{53} + 17 q^{57} + 22 q^{59} - 2 q^{61} + 12 q^{63} - 10 q^{67} + 4 q^{69} - 14 q^{71} - 3 q^{73} + 15 q^{75} + 3 q^{77} - 16 q^{79} + 38 q^{81} - q^{83} + 20 q^{87} - 3 q^{91} - 16 q^{93} + 20 q^{97} - 5 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 3 * q^7 + 5 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 7 * q^19 - 2 * q^21 - 7 * q^23 - 10 * q^25 - 18 * q^27 + 4 * q^29 + 2 * q^31 + 3 * q^33 - 4 * q^37 - 3 * q^39 + 7 * q^41 - 6 * q^43 - 12 * q^47 - 3 * q^49 - 10 * q^51 - 13 * q^53 + 17 * q^57 + 22 * q^59 - 2 * q^61 + 12 * q^63 - 10 * q^67 + 4 * q^69 - 14 * q^71 - 3 * q^73 + 15 * q^75 + 3 * q^77 - 16 * q^79 + 38 * q^81 - q^83 + 20 * q^87 - 3 * q^91 - 16 * q^93 + 20 * q^97 - 5 * q^99

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