LMFDB - Embedded newform 4544.2.a.bf.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4544,2,Mod(1,4544)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4544, 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("4544.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4544 = 2^{6} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4544.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:	$36.2840226785$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2373841.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 15x^{2} + 19x - 28 $ x^5 - 2x^4 - 9x^3 + 15x^2 + 19x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 568)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.96105$ of defining polynomial
Character	$\chi$	$=$	4544.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96105 q^{3} -0.378950 q^{5} -1.65325 q^{7} +5.76781 q^{9} +O(q^{10})$	$q+2.96105 q^{3} -0.378950 q^{5} -1.65325 q^{7} +5.76781 q^{9} -0.346752 q^{13} -1.12209 q^{15} -7.53562 q^{17} -4.76781 q^{19} -4.89535 q^{21} -7.88237 q^{23} -4.85640 q^{25} +8.19562 q^{27} -3.93430 q^{29} +4.89535 q^{31} +0.626498 q^{35} -2.49896 q^{37} -1.02675 q^{39} +8.92002 q^{41} -5.88990 q^{43} -2.18571 q^{45} +4.64027 q^{47} -4.26677 q^{49} -22.3133 q^{51} -1.62858 q^{53} -14.1177 q^{57} -2.93507 q^{59} -7.85770 q^{61} -9.53562 q^{63} +0.131402 q^{65} -2.24418 q^{67} -23.3401 q^{69} +1.00000 q^{71} +4.88782 q^{73} -14.3800 q^{75} +8.67247 q^{79} +6.96421 q^{81} -2.80754 q^{83} +2.85562 q^{85} -11.6497 q^{87} +3.89612 q^{89} +0.573267 q^{91} +14.4954 q^{93} +1.80676 q^{95} +12.6354 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

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
$3$	2.96105	1.70956	0.854781	−	0.518989i	$-0.173691\pi$
$3$	2.96105	1.70956	0.854781	+	0.518989i	$0.173691\pi$
$4$	0	0
$5$	−0.378950	−0.169472	−0.0847358	−	0.996403i	$-0.527005\pi$
$5$	−0.378950	−0.169472	−0.0847358	+	0.996403i	$0.527005\pi$
$6$	0	0
$7$	−1.65325	−0.624869	−0.312434	−	0.949939i	$-0.601144\pi$
$7$	−1.65325	−0.624869	−0.312434	+	0.949939i	$0.601144\pi$
$8$	0	0
$9$	5.76781	1.92260
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−0.346752	−0.0961717	−0.0480859	−	0.998843i	$-0.515312\pi$
$13$	−0.346752	−0.0961717	−0.0480859	+	0.998843i	$0.515312\pi$
$14$	0	0
$15$	−1.12209	−0.289722
$16$	0	0
$17$	−7.53562	−1.82766	−0.913828	−	0.406101i	$-0.866888\pi$
$17$	−7.53562	−1.82766	−0.913828	+	0.406101i	$0.866888\pi$
$18$	0	0
$19$	−4.76781	−1.09381	−0.546905	−	0.837194i	$-0.684194\pi$
$19$	−4.76781	−1.09381	−0.546905	+	0.837194i	$0.684194\pi$
$20$	0	0
$21$	−4.89535	−1.06825
$22$	0	0
$23$	−7.88237	−1.64359	−0.821794	−	0.569784i	$-0.807027\pi$
$23$	−7.88237	−1.64359	−0.821794	+	0.569784i	$0.807027\pi$
$24$	0	0
$25$	−4.85640	−0.971279
$26$	0	0
$27$	8.19562	1.57725
$28$	0	0
$29$	−3.93430	−0.730581	−0.365291	−	0.930894i	$-0.619030\pi$
$29$	−3.93430	−0.730581	−0.365291	+	0.930894i	$0.619030\pi$
$30$	0	0
$31$	4.89535	0.879230	0.439615	−	0.898186i	$-0.355115\pi$
$31$	4.89535	0.879230	0.439615	+	0.898186i	$0.355115\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.626498	0.105898
$36$	0	0
$37$	−2.49896	−0.410827	−0.205413	−	0.978675i	$-0.565854\pi$
$37$	−2.49896	−0.410827	−0.205413	+	0.978675i	$0.565854\pi$
$38$	0	0
$39$	−1.02675	−0.164412
$40$	0	0
$41$	8.92002	1.39307	0.696536	−	0.717521i	$-0.254723\pi$
$41$	8.92002	1.39307	0.696536	+	0.717521i	$0.254723\pi$
$42$	0	0
$43$	−5.88990	−0.898201	−0.449101	−	0.893481i	$-0.648256\pi$
$43$	−5.88990	−0.898201	−0.449101	+	0.893481i	$0.648256\pi$
$44$	0	0
$45$	−2.18571	−0.325827
$46$	0	0
$47$	4.64027	0.676853	0.338427	−	0.940993i	$-0.390105\pi$
$47$	4.64027	0.676853	0.338427	+	0.940993i	$0.390105\pi$
$48$	0	0
$49$	−4.26677	−0.609539
$50$	0	0
$51$	−22.3133	−3.12449
$52$	0	0
$53$	−1.62858	−0.223702	−0.111851	−	0.993725i	$-0.535678\pi$
$53$	−1.62858	−0.223702	−0.111851	+	0.993725i	$0.535678\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.1177	−1.86994
$58$	0	0
$59$	−2.93507	−0.382114	−0.191057	−	0.981579i	$-0.561192\pi$
$59$	−2.93507	−0.382114	−0.191057	+	0.981579i	$0.561192\pi$
$60$	0	0
$61$	−7.85770	−1.00608	−0.503038	−	0.864264i	$-0.667784\pi$
$61$	−7.85770	−1.00608	−0.503038	+	0.864264i	$0.667784\pi$
$62$	0	0
$63$	−9.53562	−1.20138
$64$	0	0
$65$	0.131402	0.0162984
$66$	0	0
$67$	−2.24418	−0.274170	−0.137085	−	0.990559i	$-0.543773\pi$
$67$	−2.24418	−0.274170	−0.137085	+	0.990559i	$0.543773\pi$
$68$	0	0
$69$	−23.3401	−2.80982
$70$	0	0
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	4.88782	0.572076	0.286038	−	0.958218i	$-0.407662\pi$
$73$	4.88782	0.572076	0.286038	+	0.958218i	$0.407662\pi$
$74$	0	0
$75$	−14.3800	−1.66046
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.67247	0.975729	0.487865	−	0.872919i	$-0.337776\pi$
$79$	8.67247	0.975729	0.487865	+	0.872919i	$0.337776\pi$
$80$	0	0
$81$	6.96421	0.773801
$82$	0	0
$83$	−2.80754	−0.308167	−0.154084	−	0.988058i	$-0.549242\pi$
$83$	−2.80754	−0.308167	−0.154084	+	0.988058i	$0.549242\pi$
$84$	0	0
$85$	2.85562	0.309736
$86$	0	0
$87$	−11.6497	−1.24897
$88$	0	0
$89$	3.89612	0.412988	0.206494	−	0.978448i	$-0.433795\pi$
$89$	3.89612	0.412988	0.206494	+	0.978448i	$0.433795\pi$
$90$	0	0
$91$	0.573267	0.0600947
$92$	0	0
$93$	14.4954	1.50310
$94$	0	0
$95$	1.80676	0.185370
$96$	0	0
$97$	12.6354	1.28293	0.641466	−	0.767151i	$-0.278326\pi$
$97$	12.6354	1.28293	0.641466	+	0.767151i	$0.278326\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.98524	0.396546	0.198273	−	0.980147i	$-0.436467\pi$
$101$	3.98524	0.396546	0.198273	+	0.980147i	$0.436467\pi$
$102$	0	0
$103$	−9.45902	−0.932025	−0.466013	−	0.884778i	$-0.654310\pi$
$103$	−9.45902	−0.932025	−0.466013	+	0.884778i	$0.654310\pi$
$104$	0	0
$105$	1.85509	0.181038
$106$	0	0
$107$	−14.6268	−1.41402	−0.707011	−	0.707202i	$-0.749957\pi$
$107$	−14.6268	−1.41402	−0.707011	+	0.707202i	$0.749957\pi$
$108$	0	0
$109$	9.77695	0.936462	0.468231	−	0.883606i	$-0.344892\pi$
$109$	9.77695	0.936462	0.468231	+	0.883606i	$0.344892\pi$
$110$	0	0
$111$	−7.39954	−0.702334
$112$	0	0
$113$	20.4447	1.92328	0.961640	−	0.274315i	$-0.0884511\pi$
$113$	20.4447	1.92328	0.961640	+	0.274315i	$0.0884511\pi$
$114$	0	0
$115$	2.98702	0.278542
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	12.4583	1.14205
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	26.4126	2.38155
$124$	0	0
$125$	3.73508	0.334076
$126$	0	0
$127$	15.7128	1.39428	0.697142	−	0.716933i	$-0.254454\pi$
$127$	15.7128	1.39428	0.697142	+	0.716933i	$0.254454\pi$
$128$	0	0
$129$	−17.4403	−1.53553
$130$	0	0
$131$	9.13200	0.797867	0.398933	−	0.916980i	$-0.369381\pi$
$131$	9.13200	0.797867	0.398933	+	0.916980i	$0.369381\pi$
$132$	0	0
$133$	7.88237	0.683488
$134$	0	0
$135$	−3.10573	−0.267299
$136$	0	0
$137$	−4.17717	−0.356880	−0.178440	−	0.983951i	$-0.557105\pi$
$137$	−4.17717	−0.356880	−0.178440	+	0.983951i	$0.557105\pi$
$138$	0	0
$139$	2.09772	0.177927	0.0889633	−	0.996035i	$-0.471645\pi$
$139$	2.09772	0.177927	0.0889633	+	0.996035i	$0.471645\pi$
$140$	0	0
$141$	13.7401	1.15712
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.49090	0.123813
$146$	0	0
$147$	−12.6341	−1.04204
$148$	0	0
$149$	−12.5104	−1.02489	−0.512447	−	0.858719i	$-0.671261\pi$
$149$	−12.5104	−1.02489	−0.512447	+	0.858719i	$0.671261\pi$
$150$	0	0
$151$	12.2278	0.995086	0.497543	−	0.867439i	$-0.334236\pi$
$151$	12.2278	0.995086	0.497543	+	0.867439i	$0.334236\pi$
$152$	0	0
$153$	−43.4640	−3.51386
$154$	0	0
$155$	−1.85509	−0.149005
$156$	0	0
$157$	−5.23925	−0.418137	−0.209069	−	0.977901i	$-0.567043\pi$
$157$	−5.23925	−0.418137	−0.209069	+	0.977901i	$0.567043\pi$
$158$	0	0
$159$	−4.82230	−0.382433
$160$	0	0
$161$	13.0315	1.02703
$162$	0	0
$163$	−10.4691	−0.820007	−0.410003	−	0.912084i	$-0.634472\pi$
$163$	−10.4691	−0.820007	−0.410003	+	0.912084i	$0.634472\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.3074	1.10714	0.553568	−	0.832804i	$-0.313266\pi$
$167$	14.3074	1.10714	0.553568	+	0.832804i	$0.313266\pi$
$168$	0	0
$169$	−12.8798	−0.990751
$170$	0	0
$171$	−27.4998	−2.10296
$172$	0	0
$173$	2.70467	0.205632	0.102816	−	0.994700i	$-0.467215\pi$
$173$	2.70467	0.205632	0.102816	+	0.994700i	$0.467215\pi$
$174$	0	0
$175$	8.02883	0.606922
$176$	0	0
$177$	−8.69089	−0.653248
$178$	0	0
$179$	17.9392	1.34084	0.670421	−	0.741981i	$-0.266114\pi$
$179$	17.9392	1.34084	0.670421	+	0.741981i	$0.266114\pi$
$180$	0	0
$181$	−19.1754	−1.42529	−0.712647	−	0.701523i	$-0.752504\pi$
$181$	−19.1754	−1.42529	−0.712647	+	0.701523i	$0.752504\pi$
$182$	0	0
$183$	−23.2670	−1.71995
$184$	0	0
$185$	0.946981	0.0696234
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−13.5494	−0.985574
$190$	0	0
$191$	−18.0211	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$191$	−18.0211	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$192$	0	0
$193$	−12.2935	−0.884907	−0.442453	−	0.896791i	$-0.645892\pi$
$193$	−12.2935	−0.884907	−0.442453	+	0.896791i	$0.645892\pi$
$194$	0	0
$195$	0.389087	0.0278631
$196$	0	0
$197$	21.4045	1.52501	0.762503	−	0.646984i	$-0.223970\pi$
$197$	21.4045	1.52501	0.762503	+	0.646984i	$0.223970\pi$
$198$	0	0
$199$	7.55038	0.535232	0.267616	−	0.963526i	$-0.413764\pi$
$199$	7.55038	0.535232	0.267616	+	0.963526i	$0.413764\pi$
$200$	0	0
$201$	−6.64512	−0.468711
$202$	0	0
$203$	6.50437	0.456517
$204$	0	0
$205$	−3.38024	−0.236086
$206$	0	0
$207$	−45.4640	−3.15997
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.55067	0.382124	0.191062	−	0.981578i	$-0.438807\pi$
$211$	5.55067	0.382124	0.191062	+	0.981578i	$0.438807\pi$
$212$	0	0
$213$	2.96105	0.202888
$214$	0	0
$215$	2.23198	0.152220
$216$	0	0
$217$	−8.09322	−0.549404
$218$	0	0
$219$	14.4731	0.978000
$220$	0	0
$221$	2.61299	0.175769
$222$	0	0
$223$	24.1079	1.61438	0.807192	−	0.590289i	$-0.200986\pi$
$223$	24.1079	1.61438	0.807192	+	0.590289i	$0.200986\pi$
$224$	0	0
$225$	−28.0108	−1.86739
$226$	0	0
$227$	28.8421	1.91432	0.957159	−	0.289562i	$-0.0935096\pi$
$227$	28.8421	1.91432	0.957159	+	0.289562i	$0.0935096\pi$
$228$	0	0
$229$	−24.7458	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$229$	−24.7458	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.3034	−1.06807	−0.534037	−	0.845461i	$-0.679326\pi$
$233$	−16.3034	−1.06807	−0.534037	+	0.845461i	$0.679326\pi$
$234$	0	0
$235$	−1.75843	−0.114707
$236$	0	0
$237$	25.6796	1.66807
$238$	0	0
$239$	−19.1889	−1.24122	−0.620612	−	0.784118i	$-0.713116\pi$
$239$	−19.1889	−1.24122	−0.620612	+	0.784118i	$0.713116\pi$
$240$	0	0
$241$	−17.9849	−1.15851	−0.579256	−	0.815146i	$-0.696657\pi$
$241$	−17.9849	−1.15851	−0.579256	+	0.815146i	$0.696657\pi$
$242$	0	0
$243$	−3.96551	−0.254388
$244$	0	0
$245$	1.61689	0.103299
$246$	0	0
$247$	1.65325	0.105194
$248$	0	0
$249$	−8.31325	−0.526831
$250$	0	0
$251$	10.3829	0.655362	0.327681	−	0.944788i	$-0.393733\pi$
$251$	10.3829	0.655362	0.327681	+	0.944788i	$0.393733\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.45564	0.529513
$256$	0	0
$257$	23.3154	1.45438	0.727188	−	0.686438i	$-0.240827\pi$
$257$	23.3154	1.45438	0.727188	+	0.686438i	$0.240827\pi$
$258$	0	0
$259$	4.13140	0.256713
$260$	0	0
$261$	−22.6923	−1.40462
$262$	0	0
$263$	−19.7738	−1.21931	−0.609653	−	0.792668i	$-0.708691\pi$
$263$	−19.7738	−1.21931	−0.609653	+	0.792668i	$0.708691\pi$
$264$	0	0
$265$	0.617149	0.0379112
$266$	0	0
$267$	11.5366	0.706029
$268$	0	0
$269$	−25.9666	−1.58321	−0.791605	−	0.611033i	$-0.790754\pi$
$269$	−25.9666	−1.58321	−0.791605	+	0.611033i	$0.790754\pi$
$270$	0	0
$271$	−18.4768	−1.12238	−0.561192	−	0.827686i	$-0.689657\pi$
$271$	−18.4768	−1.12238	−0.561192	+	0.827686i	$0.689657\pi$
$272$	0	0
$273$	1.69747	0.102736
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−32.5428	−1.95530	−0.977652	−	0.210228i	$-0.932580\pi$
$277$	−32.5428	−1.95530	−0.977652	+	0.210228i	$0.932580\pi$
$278$	0	0
$279$	28.2354	1.69041
$280$	0	0
$281$	−19.0603	−1.13704	−0.568522	−	0.822668i	$-0.692485\pi$
$281$	−19.0603	−1.13704	−0.568522	+	0.822668i	$0.692485\pi$
$282$	0	0
$283$	13.8900	0.825673	0.412836	−	0.910805i	$-0.364538\pi$
$283$	13.8900	0.825673	0.412836	+	0.910805i	$0.364538\pi$
$284$	0	0
$285$	5.34991	0.316901
$286$	0	0
$287$	−14.7470	−0.870488
$288$	0	0
$289$	39.7856	2.34033
$290$	0	0
$291$	37.4141	2.19325
$292$	0	0
$293$	−24.4572	−1.42880	−0.714402	−	0.699735i	$-0.753301\pi$
$293$	−24.4572	−1.42880	−0.714402	+	0.699735i	$0.753301\pi$
$294$	0	0
$295$	1.11225	0.0647575
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.73323	0.158067
$300$	0	0
$301$	9.73746	0.561258
$302$	0	0
$303$	11.8005	0.677920
$304$	0	0
$305$	2.97768	0.170501
$306$	0	0
$307$	3.80420	0.217117	0.108559	−	0.994090i	$-0.465376\pi$
$307$	3.80420	0.217117	0.108559	+	0.994090i	$0.465376\pi$
$308$	0	0
$309$	−28.0086	−1.59336
$310$	0	0
$311$	28.4622	1.61395	0.806973	−	0.590588i	$-0.201104\pi$
$311$	28.4622	1.61395	0.806973	+	0.590588i	$0.201104\pi$
$312$	0	0
$313$	22.9710	1.29840	0.649198	−	0.760619i	$-0.275104\pi$
$313$	22.9710	1.29840	0.649198	+	0.760619i	$0.275104\pi$
$314$	0	0
$315$	3.61352	0.203599
$316$	0	0
$317$	20.9398	1.17610	0.588049	−	0.808825i	$-0.299896\pi$
$317$	20.9398	1.17610	0.588049	+	0.808825i	$0.299896\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−43.3106	−2.41736
$322$	0	0
$323$	35.9284	1.99911
$324$	0	0
$325$	1.68397	0.0934096
$326$	0	0
$327$	28.9500	1.60094
$328$	0	0
$329$	−7.67152	−0.422945
$330$	0	0
$331$	−1.79528	−0.0986774	−0.0493387	−	0.998782i	$-0.515711\pi$
$331$	−1.79528	−0.0986774	−0.0493387	+	0.998782i	$0.515711\pi$
$332$	0	0
$333$	−14.4135	−0.789857
$334$	0	0
$335$	0.850431	0.0464640
$336$	0	0
$337$	−5.27697	−0.287455	−0.143728	−	0.989617i	$-0.545909\pi$
$337$	−5.27697	−0.287455	−0.143728	+	0.989617i	$0.545909\pi$
$338$	0	0
$339$	60.5379	3.28797
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.6268	1.00575
$344$	0	0
$345$	8.84473	0.476184
$346$	0	0
$347$	−28.3841	−1.52374	−0.761868	−	0.647732i	$-0.775717\pi$
$347$	−28.3841	−1.52374	−0.761868	+	0.647732i	$0.775717\pi$
$348$	0	0
$349$	−19.3462	−1.03558	−0.517790	−	0.855508i	$-0.673245\pi$
$349$	−19.3462	−1.03558	−0.517790	+	0.855508i	$0.673245\pi$
$350$	0	0
$351$	−2.84185	−0.151687
$352$	0	0
$353$	−0.858233	−0.0456791	−0.0228396	−	0.999739i	$-0.507271\pi$
$353$	−0.858233	−0.0456791	−0.0228396	+	0.999739i	$0.507271\pi$
$354$	0	0
$355$	−0.378950	−0.0201126
$356$	0	0
$357$	36.8895	1.95240
$358$	0	0
$359$	−27.2114	−1.43616	−0.718081	−	0.695960i	$-0.754979\pi$
$359$	−27.2114	−1.43616	−0.718081	+	0.695960i	$0.754979\pi$
$360$	0	0
$361$	3.73202	0.196422
$362$	0	0
$363$	−32.5715	−1.70956
$364$	0	0
$365$	−1.85224	−0.0969507
$366$	0	0
$367$	15.8232	0.825962	0.412981	−	0.910740i	$-0.364488\pi$
$367$	15.8232	0.825962	0.412981	+	0.910740i	$0.364488\pi$
$368$	0	0
$369$	51.4490	2.67833
$370$	0	0
$371$	2.69244	0.139785
$372$	0	0
$373$	−18.9005	−0.978633	−0.489316	−	0.872106i	$-0.662754\pi$
$373$	−18.9005	−0.978633	−0.489316	+	0.872106i	$0.662754\pi$
$374$	0	0
$375$	11.0598	0.571123
$376$	0	0
$377$	1.36423	0.0702612
$378$	0	0
$379$	−14.7120	−0.755706	−0.377853	−	0.925866i	$-0.623337\pi$
$379$	−14.7120	−0.755706	−0.377853	+	0.925866i	$0.623337\pi$
$380$	0	0
$381$	46.5264	2.38362
$382$	0	0
$383$	−0.617842	−0.0315702	−0.0157851	−	0.999875i	$-0.505025\pi$
$383$	−0.617842	−0.0315702	−0.0157851	+	0.999875i	$0.505025\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−33.9718	−1.72688
$388$	0	0
$389$	−31.5371	−1.59899	−0.799497	−	0.600670i	$-0.794900\pi$
$389$	−31.5371	−1.59899	−0.799497	+	0.600670i	$0.794900\pi$
$390$	0	0
$391$	59.3986	3.00392
$392$	0	0
$393$	27.0403	1.36400
$394$	0	0
$395$	−3.28643	−0.165358
$396$	0	0
$397$	−20.6036	−1.03407	−0.517034	−	0.855965i	$-0.672964\pi$
$397$	−20.6036	−1.03407	−0.517034	+	0.855965i	$0.672964\pi$
$398$	0	0
$399$	23.3401	1.16847
$400$	0	0
$401$	−13.6183	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$401$	−13.6183	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$402$	0	0
$403$	−1.69747	−0.0845571
$404$	0	0
$405$	−2.63909	−0.131137
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.8651	−1.62507	−0.812536	−	0.582911i	$-0.801914\pi$
$409$	−32.8651	−1.62507	−0.812536	+	0.582911i	$0.801914\pi$
$410$	0	0
$411$	−12.3688	−0.610109
$412$	0	0
$413$	4.85240	0.238771
$414$	0	0
$415$	1.06392	0.0522256
$416$	0	0
$417$	6.21146	0.304177
$418$	0	0
$419$	−6.97482	−0.340742	−0.170371	−	0.985380i	$-0.554497\pi$
$419$	−6.97482	−0.340742	−0.170371	+	0.985380i	$0.554497\pi$
$420$	0	0
$421$	30.7994	1.50107	0.750534	−	0.660831i	$-0.229796\pi$
$421$	30.7994	1.50107	0.750534	+	0.660831i	$0.229796\pi$
$422$	0	0
$423$	26.7642	1.30132
$424$	0	0
$425$	36.5960	1.77517
$426$	0	0
$427$	12.9907	0.628665
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.28966	0.302962	0.151481	−	0.988460i	$-0.451596\pi$
$431$	6.28966	0.302962	0.151481	+	0.988460i	$0.451596\pi$
$432$	0	0
$433$	10.4733	0.503315	0.251657	−	0.967816i	$-0.419024\pi$
$433$	10.4733	0.503315	0.251657	+	0.967816i	$0.419024\pi$
$434$	0	0
$435$	4.41464	0.211666
$436$	0	0
$437$	37.5817	1.79777
$438$	0	0
$439$	−1.99584	−0.0952564	−0.0476282	−	0.998865i	$-0.515166\pi$
$439$	−1.99584	−0.0952564	−0.0476282	+	0.998865i	$0.515166\pi$
$440$	0	0
$441$	−24.6099	−1.17190
$442$	0	0
$443$	10.6623	0.506583	0.253291	−	0.967390i	$-0.418487\pi$
$443$	10.6623	0.506583	0.253291	+	0.967390i	$0.418487\pi$
$444$	0	0
$445$	−1.47644	−0.0699897
$446$	0	0
$447$	−37.0440	−1.75212
$448$	0	0
$449$	7.29560	0.344301	0.172150	−	0.985071i	$-0.444929\pi$
$449$	7.29560	0.344301	0.172150	+	0.985071i	$0.444929\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	36.2072	1.70116
$454$	0	0
$455$	−0.217240	−0.0101843
$456$	0	0
$457$	12.2141	0.571350	0.285675	−	0.958327i	$-0.407782\pi$
$457$	12.2141	0.571350	0.285675	+	0.958327i	$0.407782\pi$
$458$	0	0
$459$	−61.7591	−2.88267
$460$	0	0
$461$	−22.2441	−1.03601	−0.518005	−	0.855377i	$-0.673325\pi$
$461$	−22.2441	−1.03601	−0.518005	+	0.855377i	$0.673325\pi$
$462$	0	0
$463$	11.7260	0.544954	0.272477	−	0.962162i	$-0.412157\pi$
$463$	11.7260	0.544954	0.272477	+	0.962162i	$0.412157\pi$
$464$	0	0
$465$	−5.49302	−0.254733
$466$	0	0
$467$	29.5507	1.36744	0.683721	−	0.729743i	$-0.260361\pi$
$467$	29.5507	1.36744	0.683721	+	0.729743i	$0.260361\pi$
$468$	0	0
$469$	3.71018	0.171320
$470$	0	0
$471$	−15.5137	−0.714832
$472$	0	0
$473$	0	0
$474$	0	0
$475$	23.1544	1.06240
$476$	0	0
$477$	−9.39332	−0.430091
$478$	0	0
$479$	−28.5858	−1.30612	−0.653058	−	0.757308i	$-0.726514\pi$
$479$	−28.5858	−1.30612	−0.653058	+	0.757308i	$0.726514\pi$
$480$	0	0
$481$	0.866520	0.0395099
$482$	0	0
$483$	38.5870	1.75577
$484$	0	0
$485$	−4.78819	−0.217421
$486$	0	0
$487$	−32.5279	−1.47398	−0.736990	−	0.675904i	$-0.763753\pi$
$487$	−32.5279	−1.47398	−0.736990	+	0.675904i	$0.763753\pi$
$488$	0	0
$489$	−30.9997	−1.40185
$490$	0	0
$491$	−10.8650	−0.490329	−0.245165	−	0.969481i	$-0.578842\pi$
$491$	−10.8650	−0.490329	−0.245165	+	0.969481i	$0.578842\pi$
$492$	0	0
$493$	29.6474	1.33525
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.65325	−0.0741583
$498$	0	0
$499$	−35.9544	−1.60954	−0.804770	−	0.593587i	$-0.797711\pi$
$499$	−35.9544	−1.60954	−0.804770	+	0.593587i	$0.797711\pi$
$500$	0	0
$501$	42.3648	1.89272
$502$	0	0
$503$	27.4029	1.22184	0.610918	−	0.791694i	$-0.290801\pi$
$503$	27.4029	1.22184	0.610918	+	0.791694i	$0.290801\pi$
$504$	0	0
$505$	−1.51021	−0.0672033
$506$	0	0
$507$	−38.1376	−1.69375
$508$	0	0
$509$	−12.0753	−0.535228	−0.267614	−	0.963526i	$-0.586235\pi$
$509$	−12.0753	−0.535228	−0.267614	+	0.963526i	$0.586235\pi$
$510$	0	0
$511$	−8.08078	−0.357473
$512$	0	0
$513$	−39.0752	−1.72521
$514$	0	0
$515$	3.58450	0.157952
$516$	0	0
$517$	0	0
$518$	0	0
$519$	8.00866	0.351541
$520$	0	0
$521$	−4.45840	−0.195326	−0.0976630	−	0.995220i	$-0.531137\pi$
$521$	−4.45840	−0.195326	−0.0976630	+	0.995220i	$0.531137\pi$
$522$	0	0
$523$	0.295068	0.0129024	0.00645122	−	0.999979i	$-0.497947\pi$
$523$	0.295068	0.0129024	0.00645122	+	0.999979i	$0.497947\pi$
$524$	0	0
$525$	23.7738	1.03757
$526$	0	0
$527$	−36.8895	−1.60693
$528$	0	0
$529$	39.1318	1.70138
$530$	0	0
$531$	−16.9289	−0.734654
$532$	0	0
$533$	−3.09303	−0.133974
$534$	0	0
$535$	5.54281	0.239637
$536$	0	0
$537$	53.1190	2.29225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.1280	1.55326	0.776632	−	0.629954i	$-0.216926\pi$
$541$	36.1280	1.55326	0.776632	+	0.629954i	$0.216926\pi$
$542$	0	0
$543$	−56.7792	−2.43663
$544$	0	0
$545$	−3.70497	−0.158704
$546$	0	0
$547$	22.8712	0.977904	0.488952	−	0.872311i	$-0.337379\pi$
$547$	22.8712	0.977904	0.488952	+	0.872311i	$0.337379\pi$
$548$	0	0
$549$	−45.3217	−1.93428
$550$	0	0
$551$	18.7580	0.799117
$552$	0	0
$553$	−14.3377	−0.609703
$554$	0	0
$555$	2.80406	0.119026
$556$	0	0
$557$	23.7811	1.00764	0.503819	−	0.863809i	$-0.331928\pi$
$557$	23.7811	1.00764	0.503819	+	0.863809i	$0.331928\pi$
$558$	0	0
$559$	2.04233	0.0863816
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.4962	0.568796	0.284398	−	0.958706i	$-0.408206\pi$
$563$	13.4962	0.568796	0.284398	+	0.958706i	$0.408206\pi$
$564$	0	0
$565$	−7.74753	−0.325941
$566$	0	0
$567$	−11.5136	−0.483524
$568$	0	0
$569$	−27.1399	−1.13776	−0.568881	−	0.822420i	$-0.692624\pi$
$569$	−27.1399	−1.13776	−0.568881	+	0.822420i	$0.692624\pi$
$570$	0	0
$571$	16.4700	0.689247	0.344624	−	0.938741i	$-0.388007\pi$
$571$	16.4700	0.689247	0.344624	+	0.938741i	$0.388007\pi$
$572$	0	0
$573$	−53.3614	−2.22921
$574$	0	0
$575$	38.2799	1.59638
$576$	0	0
$577$	−6.49910	−0.270561	−0.135281	−	0.990807i	$-0.543194\pi$
$577$	−6.49910	−0.270561	−0.135281	+	0.990807i	$0.543194\pi$
$578$	0	0
$579$	−36.4017	−1.51280
$580$	0	0
$581$	4.64155	0.192564
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0.757900	0.0313353
$586$	0	0
$587$	−1.51092	−0.0623625	−0.0311813	−	0.999514i	$-0.509927\pi$
$587$	−1.51092	−0.0623625	−0.0311813	+	0.999514i	$0.509927\pi$
$588$	0	0
$589$	−23.3401	−0.961712
$590$	0	0
$591$	63.3797	2.60709
$592$	0	0
$593$	0.562076	0.0230817	0.0115409	−	0.999933i	$-0.496326\pi$
$593$	0.562076	0.0230817	0.0115409	+	0.999933i	$0.496326\pi$
$594$	0	0
$595$	−4.72105	−0.193544
$596$	0	0
$597$	22.3570	0.915013
$598$	0	0
$599$	9.89017	0.404102	0.202051	−	0.979375i	$-0.435239\pi$
$599$	9.89017	0.404102	0.202051	+	0.979375i	$0.435239\pi$
$600$	0	0
$601$	−1.19537	−0.0487601	−0.0243800	−	0.999703i	$-0.507761\pi$
$601$	−1.19537	−0.0487601	−0.0243800	+	0.999703i	$0.507761\pi$
$602$	0	0
$603$	−12.9440	−0.527120
$604$	0	0
$605$	4.16845	0.169472
$606$	0	0
$607$	13.4562	0.546169	0.273085	−	0.961990i	$-0.411956\pi$
$607$	13.4562	0.546169	0.273085	+	0.961990i	$0.411956\pi$
$608$	0	0
$609$	19.2598	0.780445
$610$	0	0
$611$	−1.60902	−0.0650941
$612$	0	0
$613$	−21.8289	−0.881661	−0.440830	−	0.897590i	$-0.645316\pi$
$613$	−21.8289	−0.881661	−0.440830	+	0.897590i	$0.645316\pi$
$614$	0	0
$615$	−10.0091	−0.403604
$616$	0	0
$617$	7.91942	0.318824	0.159412	−	0.987212i	$-0.449040\pi$
$617$	7.91942	0.318824	0.159412	+	0.987212i	$0.449040\pi$
$618$	0	0
$619$	−9.86080	−0.396339	−0.198169	−	0.980168i	$-0.563500\pi$
$619$	−9.86080	−0.396339	−0.198169	+	0.980168i	$0.563500\pi$
$620$	0	0
$621$	−64.6010	−2.59235
$622$	0	0
$623$	−6.44126	−0.258063
$624$	0	0
$625$	22.8666	0.914663
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.8312	0.750850
$630$	0	0
$631$	−39.2791	−1.56368	−0.781839	−	0.623481i	$-0.785718\pi$
$631$	−39.2791	−1.56368	−0.781839	+	0.623481i	$0.785718\pi$
$632$	0	0
$633$	16.4358	0.653265
$634$	0	0
$635$	−5.95436	−0.236292
$636$	0	0
$637$	1.47951	0.0586204
$638$	0	0
$639$	5.76781	0.228171
$640$	0	0
$641$	−3.76883	−0.148860	−0.0744298	−	0.997226i	$-0.523714\pi$
$641$	−3.76883	−0.148860	−0.0744298	+	0.997226i	$0.523714\pi$
$642$	0	0
$643$	19.6227	0.773844	0.386922	−	0.922112i	$-0.373538\pi$
$643$	19.6227	0.773844	0.386922	+	0.922112i	$0.373538\pi$
$644$	0	0
$645$	6.60899	0.260229
$646$	0	0
$647$	3.56930	0.140324	0.0701618	−	0.997536i	$-0.477648\pi$
$647$	3.56930	0.140324	0.0701618	+	0.997536i	$0.477648\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−23.9644	−0.939240
$652$	0	0
$653$	6.69827	0.262124	0.131062	−	0.991374i	$-0.458161\pi$
$653$	6.69827	0.262124	0.131062	+	0.991374i	$0.458161\pi$
$654$	0	0
$655$	−3.46057	−0.135216
$656$	0	0
$657$	28.1920	1.09988
$658$	0	0
$659$	−44.0011	−1.71404	−0.857020	−	0.515283i	$-0.827687\pi$
$659$	−44.0011	−1.71404	−0.857020	+	0.515283i	$0.827687\pi$
$660$	0	0
$661$	9.19457	0.357627	0.178814	−	0.983883i	$-0.442774\pi$
$661$	9.19457	0.357627	0.178814	+	0.983883i	$0.442774\pi$
$662$	0	0
$663$	7.73720	0.300488
$664$	0	0
$665$	−2.98702	−0.115832
$666$	0	0
$667$	31.0116	1.20077
$668$	0	0
$669$	71.3847	2.75989
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−21.7528	−0.838508	−0.419254	−	0.907869i	$-0.637708\pi$
$673$	−21.7528	−0.838508	−0.419254	+	0.907869i	$0.637708\pi$
$674$	0	0
$675$	−39.8012	−1.53195
$676$	0	0
$677$	−25.8839	−0.994801	−0.497400	−	0.867521i	$-0.665712\pi$
$677$	−25.8839	−0.994801	−0.497400	+	0.867521i	$0.665712\pi$
$678$	0	0
$679$	−20.8895	−0.801665
$680$	0	0
$681$	85.4029	3.27265
$682$	0	0
$683$	37.1000	1.41959	0.709796	−	0.704408i	$-0.248787\pi$
$683$	37.1000	1.41959	0.709796	+	0.704408i	$0.248787\pi$
$684$	0	0
$685$	1.58294	0.0604810
$686$	0	0
$687$	−73.2735	−2.79556
$688$	0	0
$689$	0.564712	0.0215138
$690$	0	0
$691$	26.1819	0.996008	0.498004	−	0.867175i	$-0.334066\pi$
$691$	26.1819	0.996008	0.498004	+	0.867175i	$0.334066\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.794932	−0.0301535
$696$	0	0
$697$	−67.2179	−2.54606
$698$	0	0
$699$	−48.2753	−1.82594
$700$	0	0
$701$	16.4729	0.622174	0.311087	−	0.950381i	$-0.399307\pi$
$701$	16.4729	0.622174	0.311087	+	0.950381i	$0.399307\pi$
$702$	0	0
$703$	11.9146	0.449367
$704$	0	0
$705$	−5.20680	−0.196099
$706$	0	0
$707$	−6.58859	−0.247789
$708$	0	0
$709$	−25.3504	−0.952053	−0.476027	−	0.879431i	$-0.657923\pi$
$709$	−25.3504	−0.952053	−0.476027	+	0.879431i	$0.657923\pi$
$710$	0	0
$711$	50.0212	1.87594
$712$	0	0
$713$	−38.5870	−1.44509
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−56.8192	−2.12195
$718$	0	0
$719$	19.1894	0.715645	0.357822	−	0.933790i	$-0.383519\pi$
$719$	19.1894	0.715645	0.357822	+	0.933790i	$0.383519\pi$
$720$	0	0
$721$	15.6381	0.582394
$722$	0	0
$723$	−53.2543	−1.98055
$724$	0	0
$725$	19.1065	0.709598
$726$	0	0
$727$	−27.0226	−1.00221	−0.501106	−	0.865386i	$-0.667073\pi$
$727$	−27.0226	−1.00221	−0.501106	+	0.865386i	$0.667073\pi$
$728$	0	0
$729$	−32.6347	−1.20869
$730$	0	0
$731$	44.3841	1.64160
$732$	0	0
$733$	3.98521	0.147197	0.0735986	−	0.997288i	$-0.476552\pi$
$733$	3.98521	0.147197	0.0735986	+	0.997288i	$0.476552\pi$
$734$	0	0
$735$	4.78770	0.176597
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−51.9500	−1.91101	−0.955507	−	0.294969i	$-0.904691\pi$
$739$	−51.9500	−1.91101	−0.955507	+	0.294969i	$0.904691\pi$
$740$	0	0
$741$	4.89535	0.179835
$742$	0	0
$743$	9.92850	0.364241	0.182121	−	0.983276i	$-0.441704\pi$
$743$	9.92850	0.364241	0.182121	+	0.983276i	$0.441704\pi$
$744$	0	0
$745$	4.74082	0.173690
$746$	0	0
$747$	−16.1933	−0.592483
$748$	0	0
$749$	24.1817	0.883579
$750$	0	0
$751$	46.9872	1.71459	0.857294	−	0.514827i	$-0.172144\pi$
$751$	46.9872	1.71459	0.857294	+	0.514827i	$0.172144\pi$
$752$	0	0
$753$	30.7442	1.12038
$754$	0	0
$755$	−4.63373	−0.168639
$756$	0	0
$757$	−2.41270	−0.0876909	−0.0438454	−	0.999038i	$-0.513961\pi$
$757$	−2.41270	−0.0876909	−0.0438454	+	0.999038i	$0.513961\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.3566	−1.64418	−0.822088	−	0.569361i	$-0.807191\pi$
$761$	−45.3566	−1.64418	−0.822088	+	0.569361i	$0.807191\pi$
$762$	0	0
$763$	−16.1637	−0.585166
$764$	0	0
$765$	16.4707	0.595499
$766$	0	0
$767$	1.01774	0.0367486
$768$	0	0
$769$	−10.5595	−0.380785	−0.190393	−	0.981708i	$-0.560976\pi$
$769$	−10.5595	−0.380785	−0.190393	+	0.981708i	$0.560976\pi$
$770$	0	0
$771$	69.0381	2.48635
$772$	0	0
$773$	50.3563	1.81119	0.905596	−	0.424141i	$-0.139424\pi$
$773$	50.3563	1.81119	0.905596	+	0.424141i	$0.139424\pi$
$774$	0	0
$775$	−23.7738	−0.853978
$776$	0	0
$777$	12.2333	0.438867
$778$	0	0
$779$	−42.5290	−1.52376
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−32.2440	−1.15231
$784$	0	0
$785$	1.98541	0.0708624
$786$	0	0
$787$	−51.5384	−1.83715	−0.918573	−	0.395252i	$-0.870657\pi$
$787$	−51.5384	−1.83715	−0.918573	+	0.395252i	$0.870657\pi$
$788$	0	0
$789$	−58.5512	−2.08448
$790$	0	0
$791$	−33.8002	−1.20180
$792$	0	0
$793$	2.72467	0.0967560
$794$	0	0
$795$	1.82741	0.0648115
$796$	0	0
$797$	51.9750	1.84105	0.920524	−	0.390687i	$-0.127762\pi$
$797$	51.9750	1.84105	0.920524	+	0.390687i	$0.127762\pi$
$798$	0	0
$799$	−34.9673	−1.23706
$800$	0	0
$801$	22.4721	0.794012
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.93829	−0.174052
$806$	0	0
$807$	−76.8883	−2.70660
$808$	0	0
$809$	38.0402	1.33742	0.668711	−	0.743523i	$-0.266846\pi$
$809$	38.0402	1.33742	0.668711	+	0.743523i	$0.266846\pi$
$810$	0	0
$811$	−34.3417	−1.20590	−0.602949	−	0.797780i	$-0.706008\pi$
$811$	−34.3417	−1.20590	−0.602949	+	0.797780i	$0.706008\pi$
$812$	0	0
$813$	−54.7106	−1.91878
$814$	0	0
$815$	3.96728	0.138968
$816$	0	0
$817$	28.0819	0.982462
$818$	0	0
$819$	3.30650	0.115538
$820$	0	0
$821$	−5.21122	−0.181873	−0.0909365	−	0.995857i	$-0.528986\pi$
$821$	−5.21122	−0.181873	−0.0909365	+	0.995857i	$0.528986\pi$
$822$	0	0
$823$	29.4529	1.02666	0.513331	−	0.858191i	$-0.328411\pi$
$823$	29.4529	1.02666	0.513331	+	0.858191i	$0.328411\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.5932	0.785642	0.392821	−	0.919615i	$-0.371499\pi$
$827$	22.5932	0.785642	0.392821	+	0.919615i	$0.371499\pi$
$828$	0	0
$829$	−33.9811	−1.18021	−0.590106	−	0.807326i	$-0.700914\pi$
$829$	−33.9811	−1.18021	−0.590106	+	0.807326i	$0.700914\pi$
$830$	0	0
$831$	−96.3607	−3.34272
$832$	0	0
$833$	32.1528	1.11403
$834$	0	0
$835$	−5.42177	−0.187628
$836$	0	0
$837$	40.1204	1.38676
$838$	0	0
$839$	52.5584	1.81452	0.907259	−	0.420572i	$-0.138171\pi$
$839$	52.5584	1.81452	0.907259	+	0.420572i	$0.138171\pi$
$840$	0	0
$841$	−13.5213	−0.466251
$842$	0	0
$843$	−56.4386	−1.94385
$844$	0	0
$845$	4.88079	0.167904
$846$	0	0
$847$	18.1857	0.624869
$848$	0	0
$849$	41.1289	1.41154
$850$	0	0
$851$	19.6977	0.675230
$852$	0	0
$853$	−19.9042	−0.681507	−0.340753	−	0.940153i	$-0.610682\pi$
$853$	−19.9042	−0.681507	−0.340753	+	0.940153i	$0.610682\pi$
$854$	0	0
$855$	10.4211	0.356393
$856$	0	0
$857$	7.15589	0.244440	0.122220	−	0.992503i	$-0.460999\pi$
$857$	7.15589	0.244440	0.122220	+	0.992503i	$0.460999\pi$
$858$	0	0
$859$	−27.4006	−0.934896	−0.467448	−	0.884020i	$-0.654827\pi$
$859$	−27.4006	−0.934896	−0.467448	+	0.884020i	$0.654827\pi$
$860$	0	0
$861$	−43.6666	−1.48815
$862$	0	0
$863$	28.1553	0.958417	0.479209	−	0.877701i	$-0.340924\pi$
$863$	28.1553	0.958417	0.479209	+	0.877701i	$0.340924\pi$
$864$	0	0
$865$	−1.02493	−0.0348488
$866$	0	0
$867$	117.807	4.00094
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0.778174	0.0263674
$872$	0	0
$873$	72.8787	2.46657
$874$	0	0
$875$	−6.17502	−0.208754
$876$	0	0
$877$	−29.9074	−1.00990	−0.504950	−	0.863148i	$-0.668489\pi$
$877$	−29.9074	−1.00990	−0.504950	+	0.863148i	$0.668489\pi$
$878$	0	0
$879$	−72.4189	−2.44263
$880$	0	0
$881$	−35.9290	−1.21048	−0.605239	−	0.796044i	$-0.706922\pi$
$881$	−35.9290	−1.21048	−0.605239	+	0.796044i	$0.706922\pi$
$882$	0	0
$883$	11.8099	0.397435	0.198718	−	0.980057i	$-0.436322\pi$
$883$	11.8099	0.397435	0.198718	+	0.980057i	$0.436322\pi$
$884$	0	0
$885$	3.29341	0.110707
$886$	0	0
$887$	−13.6327	−0.457742	−0.228871	−	0.973457i	$-0.573503\pi$
$887$	−13.6327	−0.457742	−0.228871	+	0.973457i	$0.573503\pi$
$888$	0	0
$889$	−25.9771	−0.871245
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−22.1239	−0.740349
$894$	0	0
$895$	−6.79808	−0.227235
$896$	0	0
$897$	8.09322	0.270225
$898$	0	0
$899$	−19.2598	−0.642349
$900$	0	0
$901$	12.2723	0.408851
$902$	0	0
$903$	28.8331	0.959506
$904$	0	0
$905$	7.26650	0.241547
$906$	0	0
$907$	34.3730	1.14134	0.570668	−	0.821181i	$-0.306684\pi$
$907$	34.3730	1.14134	0.570668	+	0.821181i	$0.306684\pi$
$908$	0	0
$909$	22.9861	0.762401
$910$	0	0
$911$	9.12863	0.302445	0.151223	−	0.988500i	$-0.451679\pi$
$911$	9.12863	0.302445	0.151223	+	0.988500i	$0.451679\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	8.81704	0.291482
$916$	0	0
$917$	−15.0975	−0.498562
$918$	0	0
$919$	17.7057	0.584057	0.292028	−	0.956410i	$-0.405670\pi$
$919$	17.7057	0.584057	0.292028	+	0.956410i	$0.405670\pi$
$920$	0	0
$921$	11.2644	0.371176
$922$	0	0
$923$	−0.346752	−0.0114135
$924$	0	0
$925$	12.1359	0.399027
$926$	0	0
$927$	−54.5579	−1.79192
$928$	0	0
$929$	−13.5504	−0.444574	−0.222287	−	0.974981i	$-0.571352\pi$
$929$	−13.5504	−0.444574	−0.222287	+	0.974981i	$0.571352\pi$
$930$	0	0
$931$	20.3432	0.666720
$932$	0	0
$933$	84.2781	2.75914
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.0083	−0.392295	−0.196148	−	0.980574i	$-0.562843\pi$
$937$	−12.0083	−0.392295	−0.196148	+	0.980574i	$0.562843\pi$
$938$	0	0
$939$	68.0182	2.21969
$940$	0	0
$941$	−31.3884	−1.02323	−0.511617	−	0.859214i	$-0.670953\pi$
$941$	−31.3884	−1.02323	−0.511617	+	0.859214i	$0.670953\pi$
$942$	0	0
$943$	−70.3109	−2.28964
$944$	0	0
$945$	5.13454	0.167027
$946$	0	0
$947$	37.6655	1.22396	0.611981	−	0.790872i	$-0.290373\pi$
$947$	37.6655	1.22396	0.611981	+	0.790872i	$0.290373\pi$
$948$	0	0
$949$	−1.69486	−0.0550175
$950$	0	0
$951$	62.0039	2.01061
$952$	0	0
$953$	28.4529	0.921679	0.460840	−	0.887483i	$-0.347548\pi$
$953$	28.4529	0.921679	0.460840	+	0.887483i	$0.347548\pi$
$954$	0	0
$955$	6.82911	0.220985
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.90590	0.223003
$960$	0	0
$961$	−7.03557	−0.226954
$962$	0	0
$963$	−84.3644	−2.71861
$964$	0	0
$965$	4.65863	0.149967
$966$	0	0
$967$	50.0862	1.61066	0.805331	−	0.592825i	$-0.201988\pi$
$967$	50.0862	1.61066	0.805331	+	0.592825i	$0.201988\pi$
$968$	0	0
$969$	106.386	3.41760
$970$	0	0
$971$	42.4409	1.36199	0.680996	−	0.732287i	$-0.261547\pi$
$971$	42.4409	1.36199	0.680996	+	0.732287i	$0.261547\pi$
$972$	0	0
$973$	−3.46806	−0.111181
$974$	0	0
$975$	4.98630	0.159690
$976$	0	0
$977$	−4.12075	−0.131834	−0.0659172	−	0.997825i	$-0.520997\pi$
$977$	−4.12075	−0.131834	−0.0659172	+	0.997825i	$0.520997\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	56.3916	1.80044
$982$	0	0
$983$	−4.91038	−0.156617	−0.0783083	−	0.996929i	$-0.524952\pi$
$983$	−4.91038	−0.156617	−0.0783083	+	0.996929i	$0.524952\pi$
$984$	0	0
$985$	−8.11123	−0.258445
$986$	0	0
$987$	−22.7158	−0.723050
$988$	0	0
$989$	46.4264	1.47627
$990$	0	0
$991$	41.6586	1.32333	0.661664	−	0.749801i	$-0.269851\pi$
$991$	41.6586	1.32333	0.661664	+	0.749801i	$0.269851\pi$
$992$	0	0
$993$	−5.31591	−0.168695
$994$	0	0
$995$	−2.86122	−0.0907067
$996$	0	0
$997$	−18.7891	−0.595055	−0.297528	−	0.954713i	$-0.596162\pi$
$997$	−18.7891	−0.595055	−0.297528	+	0.954713i	$0.596162\pi$
$998$	0	0
$999$	−20.4805	−0.647976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4544.2.a.bf.1.5		5
4.3	odd	2		4544.2.a.be.1.1		5
8.3	odd	2		568.2.a.e.1.5	✓	5
8.5	even	2		1136.2.a.n.1.1		5
24.11	even	2		5112.2.a.n.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
568.2.a.e.1.5	✓	5	8.3	odd	2
1136.2.a.n.1.1		5	8.5	even	2
4544.2.a.be.1.1		5	4.3	odd	2
4544.2.a.bf.1.5		5	1.1	even	1	trivial
5112.2.a.n.1.4		5	24.11	even	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:	\( N \)	\(=\)	\( 4544 = 2^{6} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4544.a (trivial)

\(f(q)\)	\(=\)	\(q+2.96105 q^{3} -0.378950 q^{5} -1.65325 q^{7} +5.76781 q^{9} +O(q^{10})\)	\(q+2.96105 q^{3} -0.378950 q^{5} -1.65325 q^{7} +5.76781 q^{9} -0.346752 q^{13} -1.12209 q^{15} -7.53562 q^{17} -4.76781 q^{19} -4.89535 q^{21} -7.88237 q^{23} -4.85640 q^{25} +8.19562 q^{27} -3.93430 q^{29} +4.89535 q^{31} +0.626498 q^{35} -2.49896 q^{37} -1.02675 q^{39} +8.92002 q^{41} -5.88990 q^{43} -2.18571 q^{45} +4.64027 q^{47} -4.26677 q^{49} -22.3133 q^{51} -1.62858 q^{53} -14.1177 q^{57} -2.93507 q^{59} -7.85770 q^{61} -9.53562 q^{63} +0.131402 q^{65} -2.24418 q^{67} -23.3401 q^{69} +1.00000 q^{71} +4.88782 q^{73} -14.3800 q^{75} +8.67247 q^{79} +6.96421 q^{81} -2.80754 q^{83} +2.85562 q^{85} -11.6497 q^{87} +3.89612 q^{89} +0.573267 q^{91} +14.4954 q^{93} +1.80676 q^{95} +12.6354 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

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