LMFDB - Embedded newform 416.2.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(1,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.32177672409$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	416.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -3.56155 q^{5} -3.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -1.00000 q^{13} -5.56155 q^{15} +3.56155 q^{17} -6.00000 q^{19} -5.56155 q^{21} +7.68466 q^{25} -5.56155 q^{27} +8.24621 q^{29} +1.12311 q^{31} -3.12311 q^{33} +12.6847 q^{35} +2.68466 q^{37} -1.56155 q^{39} -1.12311 q^{41} -11.8078 q^{43} +2.00000 q^{45} +10.6847 q^{47} +5.68466 q^{49} +5.56155 q^{51} -13.1231 q^{53} +7.12311 q^{55} -9.36932 q^{57} +6.00000 q^{59} -11.3693 q^{61} +2.00000 q^{63} +3.56155 q^{65} -6.00000 q^{67} -10.6847 q^{71} +10.0000 q^{73} +12.0000 q^{75} +7.12311 q^{77} -12.0000 q^{79} -7.00000 q^{81} -7.36932 q^{83} -12.6847 q^{85} +12.8769 q^{87} +8.24621 q^{89} +3.56155 q^{91} +1.75379 q^{93} +21.3693 q^{95} -6.00000 q^{97} +1.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} - 7 q^{15} + 3 q^{17} - 12 q^{19} - 7 q^{21} + 3 q^{25} - 7 q^{27} - 6 q^{31} + 2 q^{33} + 13 q^{35} - 7 q^{37} + q^{39} + 6 q^{41} - 3 q^{43}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 - 7 * q^15 + 3 * q^17 - 12 * q^19 - 7 * q^21 + 3 * q^25 - 7 * q^27 - 6 * q^31 + 2 * q^33 + 13 * q^35 - 7 * q^37 + q^39 + 6 * q^41 - 3 * q^43 + 4 * q^45 + 9 * q^47 - q^49 + 7 * q^51 - 18 * q^53 + 6 * q^55 + 6 * q^57 + 12 * q^59 + 2 * q^61 + 4 * q^63 + 3 * q^65 - 12 * q^67 - 9 * q^71 + 20 * q^73 + 24 * q^75 + 6 * q^77 - 24 * q^79 - 14 * q^81 + 10 * q^83 - 13 * q^85 + 34 * q^87 + 3 * q^91 + 20 * q^93 + 18 * q^95 - 12 * q^97 - 6 * q^99

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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	−3.56155	−1.34614	−0.673070	−	0.739579i	$-0.735025\pi$
$7$	−3.56155	−1.34614	−0.673070	+	0.739579i	$0.735025\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−5.56155	−1.43599
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−5.56155	−1.21363
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−5.56155	−1.07032
$28$	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
$31$	1.12311	0.201716	0.100858	−	0.994901i	$-0.467841\pi$
$31$	1.12311	0.201716	0.100858	+	0.994901i	$0.467841\pi$
$32$	0	0
$33$	−3.12311	−0.543663
$34$	0	0
$35$	12.6847	2.14410
$36$	0	0
$37$	2.68466	0.441355	0.220678	−	0.975347i	$-0.429173\pi$
$37$	2.68466	0.441355	0.220678	+	0.975347i	$0.429173\pi$
$38$	0	0
$39$	−1.56155	−0.250049
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	−11.8078	−1.80067	−0.900334	−	0.435200i	$-0.856677\pi$
$43$	−11.8078	−1.80067	−0.900334	+	0.435200i	$0.856677\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	10.6847	1.55852	0.779259	−	0.626702i	$-0.215596\pi$
$47$	10.6847	1.55852	0.779259	+	0.626702i	$0.215596\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	−13.1231	−1.80260	−0.901299	−	0.433198i	$-0.857385\pi$
$53$	−13.1231	−1.80260	−0.901299	+	0.433198i	$0.857385\pi$
$54$	0	0
$55$	7.12311	0.960479
$56$	0	0
$57$	−9.36932	−1.24100
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−11.3693	−1.45569	−0.727846	−	0.685741i	$-0.759478\pi$
$61$	−11.3693	−1.45569	−0.727846	+	0.685741i	$0.759478\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.6847	−1.26804	−0.634018	−	0.773318i	$-0.718595\pi$
$71$	−10.6847	−1.26804	−0.634018	+	0.773318i	$0.718595\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	7.12311	0.811753
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−7.36932	−0.808888	−0.404444	−	0.914563i	$-0.632535\pi$
$83$	−7.36932	−0.808888	−0.404444	+	0.914563i	$0.632535\pi$
$84$	0	0
$85$	−12.6847	−1.37584
$86$	0	0
$87$	12.8769	1.38055
$88$	0	0
$89$	8.24621	0.874097	0.437048	−	0.899438i	$-0.356024\pi$
$89$	8.24621	0.874097	0.437048	+	0.899438i	$0.356024\pi$
$90$	0	0
$91$	3.56155	0.373352
$92$	0	0
$93$	1.75379	0.181859
$94$	0	0
$95$	21.3693	2.19245
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.12311	0.112876
$100$	0	0
$101$	1.12311	0.111753	0.0558766	−	0.998438i	$-0.482205\pi$
$101$	1.12311	0.111753	0.0558766	+	0.998438i	$0.482205\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	19.8078	1.93304
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	10.6847	1.02340	0.511702	−	0.859163i	$-0.329015\pi$
$109$	10.6847	1.02340	0.511702	+	0.859163i	$0.329015\pi$
$110$	0	0
$111$	4.19224	0.397909
$112$	0	0
$113$	−8.24621	−0.775738	−0.387869	−	0.921714i	$-0.626789\pi$
$113$	−8.24621	−0.775738	−0.387869	+	0.921714i	$0.626789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.561553	0.0519156
$118$	0	0
$119$	−12.6847	−1.16280
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.75379	−0.158134
$124$	0	0
$125$	−9.56155	−0.855211
$126$	0	0
$127$	14.2462	1.26415	0.632073	−	0.774909i	$-0.282204\pi$
$127$	14.2462	1.26415	0.632073	+	0.774909i	$0.282204\pi$
$128$	0	0
$129$	−18.4384	−1.62341
$130$	0	0
$131$	−3.31534	−0.289663	−0.144831	−	0.989456i	$-0.546264\pi$
$131$	−3.31534	−0.289663	−0.144831	+	0.989456i	$0.546264\pi$
$132$	0	0
$133$	21.3693	1.85295
$134$	0	0
$135$	19.8078	1.70478
$136$	0	0
$137$	13.1231	1.12118	0.560591	−	0.828093i	$-0.310574\pi$
$137$	13.1231	1.12118	0.560591	+	0.828093i	$0.310574\pi$
$138$	0	0
$139$	4.68466	0.397348	0.198674	−	0.980066i	$-0.436337\pi$
$139$	4.68466	0.397348	0.198674	+	0.980066i	$0.436337\pi$
$140$	0	0
$141$	16.6847	1.40510
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−29.3693	−2.43899
$146$	0	0
$147$	8.87689	0.732154
$148$	0	0
$149$	8.24621	0.675556	0.337778	−	0.941226i	$-0.390325\pi$
$149$	8.24621	0.675556	0.337778	+	0.941226i	$0.390325\pi$
$150$	0	0
$151$	−22.6847	−1.84605	−0.923026	−	0.384738i	$-0.874292\pi$
$151$	−22.6847	−1.84605	−0.923026	+	0.384738i	$0.874292\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−20.4924	−1.62515
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.3693	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$163$	15.3693	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$164$	0	0
$165$	11.1231	0.865933
$166$	0	0
$167$	−15.3693	−1.18931	−0.594657	−	0.803980i	$-0.702712\pi$
$167$	−15.3693	−1.18931	−0.594657	+	0.803980i	$0.702712\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.36932	0.257658
$172$	0	0
$173$	−3.36932	−0.256164	−0.128082	−	0.991764i	$-0.540882\pi$
$173$	−3.36932	−0.256164	−0.128082	+	0.991764i	$0.540882\pi$
$174$	0	0
$175$	−27.3693	−2.06893
$176$	0	0
$177$	9.36932	0.704241
$178$	0	0
$179$	−18.0540	−1.34942	−0.674709	−	0.738084i	$-0.735731\pi$
$179$	−18.0540	−1.34942	−0.674709	+	0.738084i	$0.735731\pi$
$180$	0	0
$181$	−7.36932	−0.547757	−0.273879	−	0.961764i	$-0.588307\pi$
$181$	−7.36932	−0.547757	−0.273879	+	0.961764i	$0.588307\pi$
$182$	0	0
$183$	−17.7538	−1.31240
$184$	0	0
$185$	−9.56155	−0.702979
$186$	0	0
$187$	−7.12311	−0.520893
$188$	0	0
$189$	19.8078	1.44080
$190$	0	0
$191$	−5.36932	−0.388510	−0.194255	−	0.980951i	$-0.562229\pi$
$191$	−5.36932	−0.388510	−0.194255	+	0.980951i	$0.562229\pi$
$192$	0	0
$193$	−23.3693	−1.68216	−0.841080	−	0.540911i	$-0.818080\pi$
$193$	−23.3693	−1.68216	−0.841080	+	0.540911i	$0.818080\pi$
$194$	0	0
$195$	5.56155	0.398271
$196$	0	0
$197$	1.31534	0.0937142	0.0468571	−	0.998902i	$-0.485079\pi$
$197$	1.31534	0.0937142	0.0468571	+	0.998902i	$0.485079\pi$
$198$	0	0
$199$	9.36932	0.664173	0.332087	−	0.943249i	$-0.392247\pi$
$199$	9.36932	0.664173	0.332087	+	0.943249i	$0.392247\pi$
$200$	0	0
$201$	−9.36932	−0.660861
$202$	0	0
$203$	−29.3693	−2.06132
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−23.8078	−1.63899	−0.819497	−	0.573083i	$-0.805747\pi$
$211$	−23.8078	−1.63899	−0.819497	+	0.573083i	$0.805747\pi$
$212$	0	0
$213$	−16.6847	−1.14321
$214$	0	0
$215$	42.0540	2.86806
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	15.6155	1.05520
$220$	0	0
$221$	−3.56155	−0.239576
$222$	0	0
$223$	−6.19224	−0.414663	−0.207331	−	0.978271i	$-0.566478\pi$
$223$	−6.19224	−0.414663	−0.207331	+	0.978271i	$0.566478\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	3.36932	0.223629	0.111815	−	0.993729i	$-0.464334\pi$
$227$	3.36932	0.223629	0.111815	+	0.993729i	$0.464334\pi$
$228$	0	0
$229$	1.31534	0.0869202	0.0434601	−	0.999055i	$-0.486162\pi$
$229$	1.31534	0.0869202	0.0434601	+	0.999055i	$0.486162\pi$
$230$	0	0
$231$	11.1231	0.731847
$232$	0	0
$233$	1.31534	0.0861709	0.0430854	−	0.999071i	$-0.486281\pi$
$233$	1.31534	0.0861709	0.0430854	+	0.999071i	$0.486281\pi$
$234$	0	0
$235$	−38.0540	−2.48237
$236$	0	0
$237$	−18.7386	−1.21721
$238$	0	0
$239$	−9.31534	−0.602559	−0.301280	−	0.953536i	$-0.597414\pi$
$239$	−9.31534	−0.602559	−0.301280	+	0.953536i	$0.597414\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−20.2462	−1.29348
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−11.5076	−0.729263
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−19.8078	−1.24041
$256$	0	0
$257$	−12.9309	−0.806605	−0.403303	−	0.915067i	$-0.632138\pi$
$257$	−12.9309	−0.806605	−0.403303	+	0.915067i	$0.632138\pi$
$258$	0	0
$259$	−9.56155	−0.594126
$260$	0	0
$261$	−4.63068	−0.286632
$262$	0	0
$263$	22.7386	1.40212	0.701062	−	0.713100i	$-0.252710\pi$
$263$	22.7386	1.40212	0.701062	+	0.713100i	$0.252710\pi$
$264$	0	0
$265$	46.7386	2.87113
$266$	0	0
$267$	12.8769	0.788053
$268$	0	0
$269$	3.36932	0.205431	0.102715	−	0.994711i	$-0.467247\pi$
$269$	3.36932	0.205431	0.102715	+	0.994711i	$0.467247\pi$
$270$	0	0
$271$	10.6847	0.649047	0.324523	−	0.945878i	$-0.394796\pi$
$271$	10.6847	0.649047	0.324523	+	0.945878i	$0.394796\pi$
$272$	0	0
$273$	5.56155	0.336600
$274$	0	0
$275$	−15.3693	−0.926805
$276$	0	0
$277$	23.3693	1.40413	0.702063	−	0.712115i	$-0.252262\pi$
$277$	23.3693	1.40413	0.702063	+	0.712115i	$0.252262\pi$
$278$	0	0
$279$	−0.630683	−0.0377580
$280$	0	0
$281$	3.75379	0.223932	0.111966	−	0.993712i	$-0.464285\pi$
$281$	3.75379	0.223932	0.111966	+	0.993712i	$0.464285\pi$
$282$	0	0
$283$	14.2462	0.846849	0.423425	−	0.905931i	$-0.360828\pi$
$283$	14.2462	0.846849	0.423425	+	0.905931i	$0.360828\pi$
$284$	0	0
$285$	33.3693	1.97663
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	−9.36932	−0.549239
$292$	0	0
$293$	−27.5616	−1.61016	−0.805082	−	0.593164i	$-0.797879\pi$
$293$	−27.5616	−1.61016	−0.805082	+	0.593164i	$0.797879\pi$
$294$	0	0
$295$	−21.3693	−1.24417
$296$	0	0
$297$	11.1231	0.645428
$298$	0	0
$299$	0	0
$300$	0	0
$301$	42.0540	2.42395
$302$	0	0
$303$	1.75379	0.100753
$304$	0	0
$305$	40.4924	2.31859
$306$	0	0
$307$	6.00000	0.342438	0.171219	−	0.985233i	$-0.445229\pi$
$307$	6.00000	0.342438	0.171219	+	0.985233i	$0.445229\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.36932	−0.531285	−0.265643	−	0.964072i	$-0.585584\pi$
$311$	−9.36932	−0.531285	−0.265643	+	0.964072i	$0.585584\pi$
$312$	0	0
$313$	−20.0540	−1.13352	−0.566759	−	0.823884i	$-0.691803\pi$
$313$	−20.0540	−1.13352	−0.566759	+	0.823884i	$0.691803\pi$
$314$	0	0
$315$	−7.12311	−0.401342
$316$	0	0
$317$	−15.7538	−0.884821	−0.442410	−	0.896813i	$-0.645877\pi$
$317$	−15.7538	−0.884821	−0.442410	+	0.896813i	$0.645877\pi$
$318$	0	0
$319$	−16.4924	−0.923398
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	−21.3693	−1.18902
$324$	0	0
$325$	−7.68466	−0.426268
$326$	0	0
$327$	16.6847	0.922664
$328$	0	0
$329$	−38.0540	−2.09798
$330$	0	0
$331$	3.36932	0.185194	0.0925972	−	0.995704i	$-0.470483\pi$
$331$	3.36932	0.185194	0.0925972	+	0.995704i	$0.470483\pi$
$332$	0	0
$333$	−1.50758	−0.0826147
$334$	0	0
$335$	21.3693	1.16753
$336$	0	0
$337$	−36.0540	−1.96399	−0.981993	−	0.188919i	$-0.939501\pi$
$337$	−36.0540	−1.96399	−0.981993	+	0.188919i	$0.939501\pi$
$338$	0	0
$339$	−12.8769	−0.699377
$340$	0	0
$341$	−2.24621	−0.121639
$342$	0	0
$343$	4.68466	0.252948
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.6847	1.53987	0.769937	−	0.638120i	$-0.220288\pi$
$347$	28.6847	1.53987	0.769937	+	0.638120i	$0.220288\pi$
$348$	0	0
$349$	−8.05398	−0.431119	−0.215560	−	0.976491i	$-0.569158\pi$
$349$	−8.05398	−0.431119	−0.215560	+	0.976491i	$0.569158\pi$
$350$	0	0
$351$	5.56155	0.296854
$352$	0	0
$353$	31.8617	1.69583	0.847915	−	0.530133i	$-0.177858\pi$
$353$	31.8617	1.69583	0.847915	+	0.530133i	$0.177858\pi$
$354$	0	0
$355$	38.0540	2.01970
$356$	0	0
$357$	−19.8078	−1.04834
$358$	0	0
$359$	−7.36932	−0.388938	−0.194469	−	0.980909i	$-0.562298\pi$
$359$	−7.36932	−0.388938	−0.194469	+	0.980909i	$0.562298\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−10.9309	−0.573722
$364$	0	0
$365$	−35.6155	−1.86420
$366$	0	0
$367$	2.63068	0.137321	0.0686603	−	0.997640i	$-0.478128\pi$
$367$	2.63068	0.137321	0.0686603	+	0.997640i	$0.478128\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	46.7386	2.42655
$372$	0	0
$373$	11.3693	0.588681	0.294340	−	0.955701i	$-0.404900\pi$
$373$	11.3693	0.588681	0.294340	+	0.955701i	$0.404900\pi$
$374$	0	0
$375$	−14.9309	−0.771027
$376$	0	0
$377$	−8.24621	−0.424701
$378$	0	0
$379$	−25.1231	−1.29049	−0.645244	−	0.763977i	$-0.723244\pi$
$379$	−25.1231	−1.29049	−0.645244	+	0.763977i	$0.723244\pi$
$380$	0	0
$381$	22.2462	1.13971
$382$	0	0
$383$	−10.6847	−0.545961	−0.272980	−	0.962020i	$-0.588009\pi$
$383$	−10.6847	−0.545961	−0.272980	+	0.962020i	$0.588009\pi$
$384$	0	0
$385$	−25.3693	−1.29294
$386$	0	0
$387$	6.63068	0.337057
$388$	0	0
$389$	3.75379	0.190325	0.0951623	−	0.995462i	$-0.469663\pi$
$389$	3.75379	0.190325	0.0951623	+	0.995462i	$0.469663\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−5.17708	−0.261149
$394$	0	0
$395$	42.7386	2.15041
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	0	0
$399$	33.3693	1.67056
$400$	0	0
$401$	22.8769	1.14242	0.571209	−	0.820805i	$-0.306475\pi$
$401$	22.8769	1.14242	0.571209	+	0.820805i	$0.306475\pi$
$402$	0	0
$403$	−1.12311	−0.0559459
$404$	0	0
$405$	24.9309	1.23882
$406$	0	0
$407$	−5.36932	−0.266147
$408$	0	0
$409$	−31.3693	−1.55111	−0.775556	−	0.631278i	$-0.782531\pi$
$409$	−31.3693	−1.55111	−0.775556	+	0.631278i	$0.782531\pi$
$410$	0	0
$411$	20.4924	1.01082
$412$	0	0
$413$	−21.3693	−1.05152
$414$	0	0
$415$	26.2462	1.28838
$416$	0	0
$417$	7.31534	0.358234
$418$	0	0
$419$	22.0540	1.07741	0.538704	−	0.842495i	$-0.318914\pi$
$419$	22.0540	1.07741	0.538704	+	0.842495i	$0.318914\pi$
$420$	0	0
$421$	−21.3153	−1.03885	−0.519423	−	0.854517i	$-0.673853\pi$
$421$	−21.3153	−1.03885	−0.519423	+	0.854517i	$0.673853\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	27.3693	1.32761
$426$	0	0
$427$	40.4924	1.95957
$428$	0	0
$429$	3.12311	0.150785
$430$	0	0
$431$	6.68466	0.321989	0.160994	−	0.986955i	$-0.448530\pi$
$431$	6.68466	0.321989	0.160994	+	0.986955i	$0.448530\pi$
$432$	0	0
$433$	26.6847	1.28238	0.641191	−	0.767381i	$-0.278440\pi$
$433$	26.6847	1.28238	0.641191	+	0.767381i	$0.278440\pi$
$434$	0	0
$435$	−45.8617	−2.19890
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.384472	0.0183498	0.00917492	−	0.999958i	$-0.497079\pi$
$439$	0.384472	0.0183498	0.00917492	+	0.999958i	$0.497079\pi$
$440$	0	0
$441$	−3.19224	−0.152011
$442$	0	0
$443$	−4.68466	−0.222575	−0.111287	−	0.993788i	$-0.535497\pi$
$443$	−4.68466	−0.222575	−0.111287	+	0.993788i	$0.535497\pi$
$444$	0	0
$445$	−29.3693	−1.39224
$446$	0	0
$447$	12.8769	0.609056
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	2.24621	0.105770
$452$	0	0
$453$	−35.4233	−1.66433
$454$	0	0
$455$	−12.6847	−0.594666
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	−41.8078	−1.94718	−0.973591	−	0.228300i	$-0.926683\pi$
$461$	−41.8078	−1.94718	−0.973591	+	0.228300i	$0.926683\pi$
$462$	0	0
$463$	17.6155	0.818663	0.409332	−	0.912386i	$-0.365762\pi$
$463$	17.6155	0.818663	0.409332	+	0.912386i	$0.365762\pi$
$464$	0	0
$465$	−6.24621	−0.289661
$466$	0	0
$467$	34.7386	1.60751	0.803756	−	0.594959i	$-0.202832\pi$
$467$	34.7386	1.60751	0.803756	+	0.594959i	$0.202832\pi$
$468$	0	0
$469$	21.3693	0.986743
$470$	0	0
$471$	3.12311	0.143905
$472$	0	0
$473$	23.6155	1.08584
$474$	0	0
$475$	−46.1080	−2.11558
$476$	0	0
$477$	7.36932	0.337418
$478$	0	0
$479$	−21.3153	−0.973923	−0.486961	−	0.873423i	$-0.661895\pi$
$479$	−21.3153	−0.973923	−0.486961	+	0.873423i	$0.661895\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.3693	0.970331
$486$	0	0
$487$	10.8769	0.492879	0.246440	−	0.969158i	$-0.420739\pi$
$487$	10.8769	0.492879	0.246440	+	0.969158i	$0.420739\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−23.3153	−1.05221	−0.526103	−	0.850421i	$-0.676347\pi$
$491$	−23.3153	−1.05221	−0.526103	+	0.850421i	$0.676347\pi$
$492$	0	0
$493$	29.3693	1.32273
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	38.0540	1.70695
$498$	0	0
$499$	−25.1231	−1.12466	−0.562332	−	0.826911i	$-0.690096\pi$
$499$	−25.1231	−1.12466	−0.562332	+	0.826911i	$0.690096\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	29.3693	1.30951	0.654757	−	0.755840i	$-0.272771\pi$
$503$	29.3693	1.30951	0.654757	+	0.755840i	$0.272771\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	1.56155	0.0693510
$508$	0	0
$509$	22.4924	0.996959	0.498480	−	0.866901i	$-0.333892\pi$
$509$	22.4924	0.996959	0.498480	+	0.866901i	$0.333892\pi$
$510$	0	0
$511$	−35.6155	−1.57554
$512$	0	0
$513$	33.3693	1.47329
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.3693	−0.939821
$518$	0	0
$519$	−5.26137	−0.230948
$520$	0	0
$521$	8.43845	0.369695	0.184848	−	0.982767i	$-0.440821\pi$
$521$	8.43845	0.369695	0.184848	+	0.982767i	$0.440821\pi$
$522$	0	0
$523$	7.50758	0.328283	0.164142	−	0.986437i	$-0.447515\pi$
$523$	7.50758	0.328283	0.164142	+	0.986437i	$0.447515\pi$
$524$	0	0
$525$	−42.7386	−1.86527
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−3.36932	−0.146216
$532$	0	0
$533$	1.12311	0.0486471
$534$	0	0
$535$	−42.7386	−1.84775
$536$	0	0
$537$	−28.1922	−1.21658
$538$	0	0
$539$	−11.3693	−0.489711
$540$	0	0
$541$	−30.6847	−1.31924	−0.659618	−	0.751601i	$-0.729282\pi$
$541$	−30.6847	−1.31924	−0.659618	+	0.751601i	$0.729282\pi$
$542$	0	0
$543$	−11.5076	−0.493837
$544$	0	0
$545$	−38.0540	−1.63005
$546$	0	0
$547$	14.0540	0.600905	0.300452	−	0.953797i	$-0.402862\pi$
$547$	14.0540	0.600905	0.300452	+	0.953797i	$0.402862\pi$
$548$	0	0
$549$	6.38447	0.272483
$550$	0	0
$551$	−49.4773	−2.10780
$552$	0	0
$553$	42.7386	1.81743
$554$	0	0
$555$	−14.9309	−0.633780
$556$	0	0
$557$	10.3002	0.436433	0.218216	−	0.975900i	$-0.429976\pi$
$557$	10.3002	0.436433	0.218216	+	0.975900i	$0.429976\pi$
$558$	0	0
$559$	11.8078	0.499415
$560$	0	0
$561$	−11.1231	−0.469618
$562$	0	0
$563$	−31.3153	−1.31978	−0.659892	−	0.751360i	$-0.729398\pi$
$563$	−31.3153	−1.31978	−0.659892	+	0.751360i	$0.729398\pi$
$564$	0	0
$565$	29.3693	1.23558
$566$	0	0
$567$	24.9309	1.04700
$568$	0	0
$569$	−13.3153	−0.558208	−0.279104	−	0.960261i	$-0.590037\pi$
$569$	−13.3153	−0.558208	−0.279104	+	0.960261i	$0.590037\pi$
$570$	0	0
$571$	21.5616	0.902323	0.451161	−	0.892442i	$-0.351010\pi$
$571$	21.5616	0.902323	0.451161	+	0.892442i	$0.351010\pi$
$572$	0	0
$573$	−8.38447	−0.350266
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.7386	1.86249	0.931247	−	0.364389i	$-0.118722\pi$
$577$	44.7386	1.86249	0.931247	+	0.364389i	$0.118722\pi$
$578$	0	0
$579$	−36.4924	−1.51657
$580$	0	0
$581$	26.2462	1.08888
$582$	0	0
$583$	26.2462	1.08701
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−15.3693	−0.634360	−0.317180	−	0.948365i	$-0.602736\pi$
$587$	−15.3693	−0.634360	−0.317180	+	0.948365i	$0.602736\pi$
$588$	0	0
$589$	−6.73863	−0.277661
$590$	0	0
$591$	2.05398	0.0844893
$592$	0	0
$593$	3.75379	0.154150	0.0770748	−	0.997025i	$-0.475442\pi$
$593$	3.75379	0.154150	0.0770748	+	0.997025i	$0.475442\pi$
$594$	0	0
$595$	45.1771	1.85208
$596$	0	0
$597$	14.6307	0.598794
$598$	0	0
$599$	−10.6307	−0.434358	−0.217179	−	0.976132i	$-0.569686\pi$
$599$	−10.6307	−0.434358	−0.217179	+	0.976132i	$0.569686\pi$
$600$	0	0
$601$	16.0540	0.654855	0.327428	−	0.944876i	$-0.393818\pi$
$601$	16.0540	0.654855	0.327428	+	0.944876i	$0.393818\pi$
$602$	0	0
$603$	3.36932	0.137209
$604$	0	0
$605$	24.9309	1.01358
$606$	0	0
$607$	25.8617	1.04970	0.524848	−	0.851196i	$-0.324122\pi$
$607$	25.8617	1.04970	0.524848	+	0.851196i	$0.324122\pi$
$608$	0	0
$609$	−45.8617	−1.85841
$610$	0	0
$611$	−10.6847	−0.432255
$612$	0	0
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	0	0
$615$	6.24621	0.251872
$616$	0	0
$617$	17.6155	0.709174	0.354587	−	0.935023i	$-0.384621\pi$
$617$	17.6155	0.709174	0.354587	+	0.935023i	$0.384621\pi$
$618$	0	0
$619$	15.3693	0.617745	0.308873	−	0.951103i	$-0.400048\pi$
$619$	15.3693	0.617745	0.308873	+	0.951103i	$0.400048\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−29.3693	−1.17666
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	18.7386	0.748349
$628$	0	0
$629$	9.56155	0.381244
$630$	0	0
$631$	−3.94602	−0.157089	−0.0785444	−	0.996911i	$-0.525027\pi$
$631$	−3.94602	−0.157089	−0.0785444	+	0.996911i	$0.525027\pi$
$632$	0	0
$633$	−37.1771	−1.47766
$634$	0	0
$635$	−50.7386	−2.01350
$636$	0	0
$637$	−5.68466	−0.225234
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−22.4924	−0.888397	−0.444199	−	0.895928i	$-0.646512\pi$
$641$	−22.4924	−0.888397	−0.444199	+	0.895928i	$0.646512\pi$
$642$	0	0
$643$	18.0000	0.709851	0.354925	−	0.934895i	$-0.384506\pi$
$643$	18.0000	0.709851	0.354925	+	0.934895i	$0.384506\pi$
$644$	0	0
$645$	65.6695	2.58573
$646$	0	0
$647$	−44.1080	−1.73406	−0.867031	−	0.498254i	$-0.833975\pi$
$647$	−44.1080	−1.73406	−0.867031	+	0.498254i	$0.833975\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−6.24621	−0.244808
$652$	0	0
$653$	−6.38447	−0.249844	−0.124922	−	0.992167i	$-0.539868\pi$
$653$	−6.38447	−0.249844	−0.124922	+	0.992167i	$0.539868\pi$
$654$	0	0
$655$	11.8078	0.461368
$656$	0	0
$657$	−5.61553	−0.219083
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	28.7386	1.11780	0.558902	−	0.829234i	$-0.311223\pi$
$661$	28.7386	1.11780	0.558902	+	0.829234i	$0.311223\pi$
$662$	0	0
$663$	−5.56155	−0.215993
$664$	0	0
$665$	−76.1080	−2.95134
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.66950	−0.373845
$670$	0	0
$671$	22.7386	0.877815
$672$	0	0
$673$	16.0540	0.618835	0.309418	−	0.950926i	$-0.399866\pi$
$673$	16.0540	0.618835	0.309418	+	0.950926i	$0.399866\pi$
$674$	0	0
$675$	−42.7386	−1.64501
$676$	0	0
$677$	−3.36932	−0.129493	−0.0647467	−	0.997902i	$-0.520624\pi$
$677$	−3.36932	−0.129493	−0.0647467	+	0.997902i	$0.520624\pi$
$678$	0	0
$679$	21.3693	0.820079
$680$	0	0
$681$	5.26137	0.201616
$682$	0	0
$683$	−19.3693	−0.741146	−0.370573	−	0.928803i	$-0.620839\pi$
$683$	−19.3693	−0.741146	−0.370573	+	0.928803i	$0.620839\pi$
$684$	0	0
$685$	−46.7386	−1.78579
$686$	0	0
$687$	2.05398	0.0783640
$688$	0	0
$689$	13.1231	0.499951
$690$	0	0
$691$	13.1231	0.499226	0.249613	−	0.968346i	$-0.419697\pi$
$691$	13.1231	0.499226	0.249613	+	0.968346i	$0.419697\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−16.6847	−0.632885
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	2.05398	0.0776885
$700$	0	0
$701$	−15.3693	−0.580491	−0.290246	−	0.956952i	$-0.593737\pi$
$701$	−15.3693	−0.580491	−0.290246	+	0.956952i	$0.593737\pi$
$702$	0	0
$703$	−16.1080	−0.607523
$704$	0	0
$705$	−59.4233	−2.23801
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	−32.7386	−1.22953	−0.614763	−	0.788712i	$-0.710748\pi$
$709$	−32.7386	−1.22953	−0.614763	+	0.788712i	$0.710748\pi$
$710$	0	0
$711$	6.73863	0.252719
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−7.12311	−0.266389
$716$	0	0
$717$	−14.5464	−0.543245
$718$	0	0
$719$	−9.36932	−0.349417	−0.174708	−	0.984620i	$-0.555898\pi$
$719$	−9.36932	−0.349417	−0.174708	+	0.984620i	$0.555898\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.12311	0.116150
$724$	0	0
$725$	63.3693	2.35348
$726$	0	0
$727$	0.384472	0.0142593	0.00712964	−	0.999975i	$-0.497731\pi$
$727$	0.384472	0.0142593	0.00712964	+	0.999975i	$0.497731\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−42.0540	−1.55542
$732$	0	0
$733$	12.0540	0.445224	0.222612	−	0.974907i	$-0.428542\pi$
$733$	12.0540	0.445224	0.222612	+	0.974907i	$0.428542\pi$
$734$	0	0
$735$	−31.6155	−1.16616
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−22.1080	−0.813254	−0.406627	−	0.913594i	$-0.633295\pi$
$739$	−22.1080	−0.813254	−0.406627	+	0.913594i	$0.633295\pi$
$740$	0	0
$741$	9.36932	0.344190
$742$	0	0
$743$	−1.31534	−0.0482552	−0.0241276	−	0.999709i	$-0.507681\pi$
$743$	−1.31534	−0.0482552	−0.0241276	+	0.999709i	$0.507681\pi$
$744$	0	0
$745$	−29.3693	−1.07601
$746$	0	0
$747$	4.13826	0.151411
$748$	0	0
$749$	−42.7386	−1.56164
$750$	0	0
$751$	9.75379	0.355921	0.177960	−	0.984038i	$-0.443050\pi$
$751$	9.75379	0.355921	0.177960	+	0.984038i	$0.443050\pi$
$752$	0	0
$753$	24.9848	0.910498
$754$	0	0
$755$	80.7926	2.94034
$756$	0	0
$757$	15.3693	0.558607	0.279304	−	0.960203i	$-0.409896\pi$
$757$	15.3693	0.558607	0.279304	+	0.960203i	$0.409896\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−38.0540	−1.37765
$764$	0	0
$765$	7.12311	0.257536
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−15.3693	−0.554232	−0.277116	−	0.960837i	$-0.589379\pi$
$769$	−15.3693	−0.554232	−0.277116	+	0.960837i	$0.589379\pi$
$770$	0	0
$771$	−20.1922	−0.727206
$772$	0	0
$773$	−13.3153	−0.478920	−0.239460	−	0.970906i	$-0.576970\pi$
$773$	−13.3153	−0.478920	−0.239460	+	0.970906i	$0.576970\pi$
$774$	0	0
$775$	8.63068	0.310023
$776$	0	0
$777$	−14.9309	−0.535642
$778$	0	0
$779$	6.73863	0.241437
$780$	0	0
$781$	21.3693	0.764654
$782$	0	0
$783$	−45.8617	−1.63896
$784$	0	0
$785$	−7.12311	−0.254235
$786$	0	0
$787$	5.61553	0.200172	0.100086	−	0.994979i	$-0.468088\pi$
$787$	5.61553	0.200172	0.100086	+	0.994979i	$0.468088\pi$
$788$	0	0
$789$	35.5076	1.26410
$790$	0	0
$791$	29.3693	1.04425
$792$	0	0
$793$	11.3693	0.403736
$794$	0	0
$795$	72.9848	2.58851
$796$	0	0
$797$	36.7386	1.30135	0.650675	−	0.759357i	$-0.274486\pi$
$797$	36.7386	1.30135	0.650675	+	0.759357i	$0.274486\pi$
$798$	0	0
$799$	38.0540	1.34625
$800$	0	0
$801$	−4.63068	−0.163617
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.26137	0.185209
$808$	0	0
$809$	−11.0691	−0.389170	−0.194585	−	0.980886i	$-0.562336\pi$
$809$	−11.0691	−0.389170	−0.194585	+	0.980886i	$0.562336\pi$
$810$	0	0
$811$	31.8617	1.11882	0.559408	−	0.828892i	$-0.311028\pi$
$811$	31.8617	1.11882	0.559408	+	0.828892i	$0.311028\pi$
$812$	0	0
$813$	16.6847	0.585157
$814$	0	0
$815$	−54.7386	−1.91741
$816$	0	0
$817$	70.8466	2.47861
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−32.0540	−1.11869	−0.559346	−	0.828934i	$-0.688948\pi$
$821$	−32.0540	−1.11869	−0.559346	+	0.828934i	$0.688948\pi$
$822$	0	0
$823$	14.6307	0.509994	0.254997	−	0.966942i	$-0.417926\pi$
$823$	14.6307	0.509994	0.254997	+	0.966942i	$0.417926\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−12.7386	−0.442966	−0.221483	−	0.975164i	$-0.571090\pi$
$827$	−12.7386	−0.442966	−0.221483	+	0.975164i	$0.571090\pi$
$828$	0	0
$829$	16.7386	0.581357	0.290678	−	0.956821i	$-0.406119\pi$
$829$	16.7386	0.581357	0.290678	+	0.956821i	$0.406119\pi$
$830$	0	0
$831$	36.4924	1.26591
$832$	0	0
$833$	20.2462	0.701490
$834$	0	0
$835$	54.7386	1.89431
$836$	0	0
$837$	−6.24621	−0.215901
$838$	0	0
$839$	38.1080	1.31563	0.657816	−	0.753178i	$-0.271480\pi$
$839$	38.1080	1.31563	0.657816	+	0.753178i	$0.271480\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	5.86174	0.201889
$844$	0	0
$845$	−3.56155	−0.122521
$846$	0	0
$847$	24.9309	0.856635
$848$	0	0
$849$	22.2462	0.763488
$850$	0	0
$851$	0	0
$852$	0	0
$853$	9.31534	0.318951	0.159476	−	0.987202i	$-0.449020\pi$
$853$	9.31534	0.318951	0.159476	+	0.987202i	$0.449020\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−15.7538	−0.538139	−0.269070	−	0.963121i	$-0.586716\pi$
$857$	−15.7538	−0.538139	−0.269070	+	0.963121i	$0.586716\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	6.24621	0.212870
$862$	0	0
$863$	−12.0540	−0.410322	−0.205161	−	0.978728i	$-0.565772\pi$
$863$	−12.0540	−0.410322	−0.205161	+	0.978728i	$0.565772\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	−6.73863	−0.228856
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	3.36932	0.114034
$874$	0	0
$875$	34.0540	1.15123
$876$	0	0
$877$	54.7926	1.85021	0.925107	−	0.379705i	$-0.123975\pi$
$877$	54.7926	1.85021	0.925107	+	0.379705i	$0.123975\pi$
$878$	0	0
$879$	−43.0388	−1.45166
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	21.5616	0.725604	0.362802	−	0.931866i	$-0.381820\pi$
$883$	21.5616	0.725604	0.362802	+	0.931866i	$0.381820\pi$
$884$	0	0
$885$	−33.3693	−1.12170
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−50.7386	−1.70172
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	−64.1080	−2.14529
$894$	0	0
$895$	64.3002	2.14932
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.26137	0.308884
$900$	0	0
$901$	−46.7386	−1.55709
$902$	0	0
$903$	65.6695	2.18534
$904$	0	0
$905$	26.2462	0.872454
$906$	0	0
$907$	35.4233	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$907$	35.4233	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$908$	0	0
$909$	−0.630683	−0.0209184
$910$	0	0
$911$	2.63068	0.0871584	0.0435792	−	0.999050i	$-0.486124\pi$
$911$	2.63068	0.0871584	0.0435792	+	0.999050i	$0.486124\pi$
$912$	0	0
$913$	14.7386	0.487778
$914$	0	0
$915$	63.2311	2.09035
$916$	0	0
$917$	11.8078	0.389927
$918$	0	0
$919$	−50.2462	−1.65747	−0.828735	−	0.559642i	$-0.810939\pi$
$919$	−50.2462	−1.65747	−0.828735	+	0.559642i	$0.810939\pi$
$920$	0	0
$921$	9.36932	0.308729
$922$	0	0
$923$	10.6847	0.351690
$924$	0	0
$925$	20.6307	0.678333
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−5.61553	−0.184240	−0.0921198	−	0.995748i	$-0.529364\pi$
$929$	−5.61553	−0.184240	−0.0921198	+	0.995748i	$0.529364\pi$
$930$	0	0
$931$	−34.1080	−1.11784
$932$	0	0
$933$	−14.6307	−0.478987
$934$	0	0
$935$	25.3693	0.829665
$936$	0	0
$937$	−8.73863	−0.285479	−0.142739	−	0.989760i	$-0.545591\pi$
$937$	−8.73863	−0.285479	−0.142739	+	0.989760i	$0.545591\pi$
$938$	0	0
$939$	−31.3153	−1.02194
$940$	0	0
$941$	25.3153	0.825257	0.412628	−	0.910900i	$-0.364611\pi$
$941$	25.3153	0.825257	0.412628	+	0.910900i	$0.364611\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−70.5464	−2.29487
$946$	0	0
$947$	−2.00000	−0.0649913	−0.0324956	−	0.999472i	$-0.510346\pi$
$947$	−2.00000	−0.0649913	−0.0324956	+	0.999472i	$0.510346\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−24.6004	−0.797722
$952$	0	0
$953$	−41.8078	−1.35429	−0.677143	−	0.735851i	$-0.736782\pi$
$953$	−41.8078	−1.35429	−0.677143	+	0.735851i	$0.736782\pi$
$954$	0	0
$955$	19.1231	0.618809
$956$	0	0
$957$	−25.7538	−0.832502
$958$	0	0
$959$	−46.7386	−1.50927
$960$	0	0
$961$	−29.7386	−0.959311
$962$	0	0
$963$	−6.73863	−0.217149
$964$	0	0
$965$	83.2311	2.67930
$966$	0	0
$967$	−53.4233	−1.71798	−0.858989	−	0.511995i	$-0.828907\pi$
$967$	−53.4233	−1.71798	−0.858989	+	0.511995i	$0.828907\pi$
$968$	0	0
$969$	−33.3693	−1.07198
$970$	0	0
$971$	3.31534	0.106394	0.0531972	−	0.998584i	$-0.483059\pi$
$971$	3.31534	0.106394	0.0531972	+	0.998584i	$0.483059\pi$
$972$	0	0
$973$	−16.6847	−0.534886
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	−14.9848	−0.479408	−0.239704	−	0.970846i	$-0.577050\pi$
$977$	−14.9848	−0.479408	−0.239704	+	0.970846i	$0.577050\pi$
$978$	0	0
$979$	−16.4924	−0.527100
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	21.4233	0.683297	0.341648	−	0.939828i	$-0.389015\pi$
$983$	21.4233	0.683297	0.341648	+	0.939828i	$0.389015\pi$
$984$	0	0
$985$	−4.68466	−0.149266
$986$	0	0
$987$	−59.4233	−1.89146
$988$	0	0
$989$	0	0
$990$	0	0
$991$	21.3693	0.678819	0.339409	−	0.940639i	$-0.389773\pi$
$991$	21.3693	0.678819	0.339409	+	0.940639i	$0.389773\pi$
$992$	0	0
$993$	5.26137	0.166964
$994$	0	0
$995$	−33.3693	−1.05788
$996$	0	0
$997$	−4.73863	−0.150074	−0.0750370	−	0.997181i	$-0.523907\pi$
$997$	−4.73863	−0.150074	−0.0750370	+	0.997181i	$0.523907\pi$
$998$	0	0
$999$	−14.9309	−0.472392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.a.c.1.2	✓	2
3.2	odd	2		3744.2.a.x.1.2		2
4.3	odd	2		416.2.a.e.1.1	yes	2
8.3	odd	2		832.2.a.l.1.2		2
8.5	even	2		832.2.a.o.1.1		2
12.11	even	2		3744.2.a.y.1.2		2
13.12	even	2		5408.2.a.p.1.2		2
16.3	odd	4		3328.2.b.u.1665.2		4
16.5	even	4		3328.2.b.ba.1665.2		4
16.11	odd	4		3328.2.b.u.1665.3		4
16.13	even	4		3328.2.b.ba.1665.3		4
24.5	odd	2		7488.2.a.ce.1.1		2
24.11	even	2		7488.2.a.cf.1.1		2
52.51	odd	2		5408.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.c.1.2	✓	2	1.1	even	1	trivial
416.2.a.e.1.1	yes	2	4.3	odd	2
832.2.a.l.1.2		2	8.3	odd	2
832.2.a.o.1.1		2	8.5	even	2
3328.2.b.u.1665.2		4	16.3	odd	4
3328.2.b.u.1665.3		4	16.11	odd	4
3328.2.b.ba.1665.2		4	16.5	even	4
3328.2.b.ba.1665.3		4	16.13	even	4
3744.2.a.x.1.2		2	3.2	odd	2
3744.2.a.y.1.2		2	12.11	even	2
5408.2.a.p.1.2		2	13.12	even	2
5408.2.a.bd.1.1		2	52.51	odd	2
7488.2.a.ce.1.1		2	24.5	odd	2
7488.2.a.cf.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -3.56155 q^{5} -3.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -1.00000 q^{13} -5.56155 q^{15} +3.56155 q^{17} -6.00000 q^{19} -5.56155 q^{21} +7.68466 q^{25} -5.56155 q^{27} +8.24621 q^{29} +1.12311 q^{31} -3.12311 q^{33} +12.6847 q^{35} +2.68466 q^{37} -1.56155 q^{39} -1.12311 q^{41} -11.8078 q^{43} +2.00000 q^{45} +10.6847 q^{47} +5.68466 q^{49} +5.56155 q^{51} -13.1231 q^{53} +7.12311 q^{55} -9.36932 q^{57} +6.00000 q^{59} -11.3693 q^{61} +2.00000 q^{63} +3.56155 q^{65} -6.00000 q^{67} -10.6847 q^{71} +10.0000 q^{73} +12.0000 q^{75} +7.12311 q^{77} -12.0000 q^{79} -7.00000 q^{81} -7.36932 q^{83} -12.6847 q^{85} +12.8769 q^{87} +8.24621 q^{89} +3.56155 q^{91} +1.75379 q^{93} +21.3693 q^{95} -6.00000 q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} - 7 q^{15} + 3 q^{17} - 12 q^{19} - 7 q^{21} + 3 q^{25} - 7 q^{27} - 6 q^{31} + 2 q^{33} + 13 q^{35} - 7 q^{37} + q^{39} + 6 q^{41} - 3 q^{43}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 - 7 * q^15 + 3 * q^17 - 12 * q^19 - 7 * q^21 + 3 * q^25 - 7 * q^27 - 6 * q^31 + 2 * q^33 + 13 * q^35 - 7 * q^37 + q^39 + 6 * q^41 - 3 * q^43 + 4 * q^45 + 9 * q^47 - q^49 + 7 * q^51 - 18 * q^53 + 6 * q^55 + 6 * q^57 + 12 * q^59 + 2 * q^61 + 4 * q^63 + 3 * q^65 - 12 * q^67 - 9 * q^71 + 20 * q^73 + 24 * q^75 + 6 * q^77 - 24 * q^79 - 14 * q^81 + 10 * q^83 - 13 * q^85 + 34 * q^87 + 3 * q^91 + 20 * q^93 + 18 * q^95 - 12 * q^97 - 6 * q^99

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