LMFDB - Embedded newform 3328.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.82843 q^{5} +3.82843 q^{7} -2.00000 q^{9} +4.82843 q^{11} -1.00000 q^{13} -3.82843 q^{15} +6.65685 q^{17} -4.00000 q^{19} -3.82843 q^{21} -3.17157 q^{23} +9.65685 q^{25} +5.00000 q^{27} +3.17157 q^{29} -4.82843 q^{33} +14.6569 q^{35} +3.82843 q^{37} +1.00000 q^{39} +2.82843 q^{41} -3.00000 q^{43} -7.65685 q^{45} -11.4853 q^{47} +7.65685 q^{49} -6.65685 q^{51} +3.17157 q^{53} +18.4853 q^{55} +4.00000 q^{57} -5.17157 q^{59} +10.8284 q^{61} -7.65685 q^{63} -3.82843 q^{65} +3.65685 q^{67} +3.17157 q^{69} -10.1716 q^{71} +5.17157 q^{73} -9.65685 q^{75} +18.4853 q^{77} -7.65685 q^{79} +1.00000 q^{81} +6.00000 q^{83} +25.4853 q^{85} -3.17157 q^{87} -17.6569 q^{89} -3.82843 q^{91} -15.3137 q^{95} -14.4853 q^{97} -9.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 2 q^{21} - 12 q^{23} + 8 q^{25} + 10 q^{27} + 12 q^{29} - 4 q^{33} + 18 q^{35} + 2 q^{37} + 2 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 + 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 2 * q^21 - 12 * q^23 + 8 * q^25 + 10 * q^27 + 12 * q^29 - 4 * q^33 + 18 * q^35 + 2 * q^37 + 2 * q^39 - 6 * q^43 - 4 * q^45 - 6 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 + 20 * q^55 + 8 * q^57 - 16 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 - 4 * q^67 + 12 * q^69 - 26 * q^71 + 16 * q^73 - 8 * q^75 + 20 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 + 34 * q^85 - 12 * q^87 - 24 * q^89 - 2 * q^91 - 8 * q^95 - 12 * q^97 - 8 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	3.82843	1.71212	0.856062	−	0.516873i	$-0.172904\pi$
$5$	3.82843	1.71212	0.856062	+	0.516873i	$0.172904\pi$
$6$	0	0
$7$	3.82843	1.44701	0.723505	−	0.690319i	$-0.242530\pi$
$7$	3.82843	1.44701	0.723505	+	0.690319i	$0.242530\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.82843	−0.988496
$16$	0	0
$17$	6.65685	1.61452	0.807262	−	0.590193i	$-0.200948\pi$
$17$	6.65685	1.61452	0.807262	+	0.590193i	$0.200948\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−3.82843	−0.835431
$22$	0	0
$23$	−3.17157	−0.661319	−0.330659	−	0.943750i	$-0.607271\pi$
$23$	−3.17157	−0.661319	−0.330659	+	0.943750i	$0.607271\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	3.17157	0.588946	0.294473	−	0.955660i	$-0.404856\pi$
$29$	3.17157	0.588946	0.294473	+	0.955660i	$0.404856\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.82843	−0.840521
$34$	0	0
$35$	14.6569	2.47746
$36$	0	0
$37$	3.82843	0.629390	0.314695	−	0.949193i	$-0.398098\pi$
$37$	3.82843	0.629390	0.314695	+	0.949193i	$0.398098\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	−7.65685	−1.14142
$46$	0	0
$47$	−11.4853	−1.67530	−0.837650	−	0.546207i	$-0.816071\pi$
$47$	−11.4853	−1.67530	−0.837650	+	0.546207i	$0.816071\pi$
$48$	0	0
$49$	7.65685	1.09384
$50$	0	0
$51$	−6.65685	−0.932146
$52$	0	0
$53$	3.17157	0.435649	0.217825	−	0.975988i	$-0.430104\pi$
$53$	3.17157	0.435649	0.217825	+	0.975988i	$0.430104\pi$
$54$	0	0
$55$	18.4853	2.49255
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−5.17157	−0.673281	−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157	−0.673281	−0.336641	+	0.941633i	$0.609291\pi$
$60$	0	0
$61$	10.8284	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$61$	10.8284	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$62$	0	0
$63$	−7.65685	−0.964673
$64$	0	0
$65$	−3.82843	−0.474858
$66$	0	0
$67$	3.65685	0.446756	0.223378	−	0.974732i	$-0.428292\pi$
$67$	3.65685	0.446756	0.223378	+	0.974732i	$0.428292\pi$
$68$	0	0
$69$	3.17157	0.381813
$70$	0	0
$71$	−10.1716	−1.20714	−0.603572	−	0.797309i	$-0.706256\pi$
$71$	−10.1716	−1.20714	−0.603572	+	0.797309i	$0.706256\pi$
$72$	0	0
$73$	5.17157	0.605287	0.302643	−	0.953104i	$-0.402131\pi$
$73$	5.17157	0.605287	0.302643	+	0.953104i	$0.402131\pi$
$74$	0	0
$75$	−9.65685	−1.11508
$76$	0	0
$77$	18.4853	2.10659
$78$	0	0
$79$	−7.65685	−0.861463	−0.430732	−	0.902480i	$-0.641744\pi$
$79$	−7.65685	−0.861463	−0.430732	+	0.902480i	$0.641744\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	25.4853	2.76427
$86$	0	0
$87$	−3.17157	−0.340028
$88$	0	0
$89$	−17.6569	−1.87162	−0.935811	−	0.352501i	$-0.885331\pi$
$89$	−17.6569	−1.87162	−0.935811	+	0.352501i	$0.885331\pi$
$90$	0	0
$91$	−3.82843	−0.401328
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−15.3137	−1.57115
$96$	0	0
$97$	−14.4853	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$97$	−14.4853	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$98$	0	0
$99$	−9.65685	−0.970550
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−14.6569	−1.43036
$106$	0	0
$107$	−15.3137	−1.48043	−0.740216	−	0.672369i	$-0.765277\pi$
$107$	−15.3137	−1.48043	−0.740216	+	0.672369i	$0.765277\pi$
$108$	0	0
$109$	3.82843	0.366697	0.183348	−	0.983048i	$-0.441306\pi$
$109$	3.82843	0.366697	0.183348	+	0.983048i	$0.441306\pi$
$110$	0	0
$111$	−3.82843	−0.363378
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−12.1421	−1.13226
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	25.4853	2.33623
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	−2.82843	−0.255031
$124$	0	0
$125$	17.8284	1.59462
$126$	0	0
$127$	18.4853	1.64030	0.820152	−	0.572146i	$-0.193889\pi$
$127$	18.4853	1.64030	0.820152	+	0.572146i	$0.193889\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	2.65685	0.232130	0.116065	−	0.993242i	$-0.462972\pi$
$131$	2.65685	0.232130	0.116065	+	0.993242i	$0.462972\pi$
$132$	0	0
$133$	−15.3137	−1.32787
$134$	0	0
$135$	19.1421	1.64749
$136$	0	0
$137$	−20.1421	−1.72086	−0.860429	−	0.509570i	$-0.829805\pi$
$137$	−20.1421	−1.72086	−0.860429	+	0.509570i	$0.829805\pi$
$138$	0	0
$139$	18.6569	1.58245	0.791227	−	0.611523i	$-0.209443\pi$
$139$	18.6569	1.58245	0.791227	+	0.611523i	$0.209443\pi$
$140$	0	0
$141$	11.4853	0.967235
$142$	0	0
$143$	−4.82843	−0.403773
$144$	0	0
$145$	12.1421	1.00835
$146$	0	0
$147$	−7.65685	−0.631527
$148$	0	0
$149$	−1.31371	−0.107623	−0.0538116	−	0.998551i	$-0.517137\pi$
$149$	−1.31371	−0.107623	−0.0538116	+	0.998551i	$0.517137\pi$
$150$	0	0
$151$	2.51472	0.204645	0.102322	−	0.994751i	$-0.467373\pi$
$151$	2.51472	0.204645	0.102322	+	0.994751i	$0.467373\pi$
$152$	0	0
$153$	−13.3137	−1.07635
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.48528	−0.357964	−0.178982	−	0.983852i	$-0.557280\pi$
$157$	−4.48528	−0.357964	−0.178982	+	0.983852i	$0.557280\pi$
$158$	0	0
$159$	−3.17157	−0.251522
$160$	0	0
$161$	−12.1421	−0.956934
$162$	0	0
$163$	−12.8284	−1.00480	−0.502400	−	0.864635i	$-0.667549\pi$
$163$	−12.8284	−1.00480	−0.502400	+	0.864635i	$0.667549\pi$
$164$	0	0
$165$	−18.4853	−1.43908
$166$	0	0
$167$	21.6569	1.67586	0.837929	−	0.545779i	$-0.183766\pi$
$167$	21.6569	1.67586	0.837929	+	0.545779i	$0.183766\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	−22.9706	−1.74642	−0.873210	−	0.487345i	$-0.837966\pi$
$173$	−22.9706	−1.74642	−0.873210	+	0.487345i	$0.837966\pi$
$174$	0	0
$175$	36.9706	2.79471
$176$	0	0
$177$	5.17157	0.388719
$178$	0	0
$179$	8.31371	0.621396	0.310698	−	0.950509i	$-0.399437\pi$
$179$	8.31371	0.621396	0.310698	+	0.950509i	$0.399437\pi$
$180$	0	0
$181$	22.9706	1.70739	0.853694	−	0.520775i	$-0.174357\pi$
$181$	22.9706	1.70739	0.853694	+	0.520775i	$0.174357\pi$
$182$	0	0
$183$	−10.8284	−0.800460
$184$	0	0
$185$	14.6569	1.07759
$186$	0	0
$187$	32.1421	2.35047
$188$	0	0
$189$	19.1421	1.39239
$190$	0	0
$191$	3.17157	0.229487	0.114743	−	0.993395i	$-0.463395\pi$
$191$	3.17157	0.229487	0.114743	+	0.993395i	$0.463395\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	3.82843	0.274159
$196$	0	0
$197$	17.8284	1.27022	0.635111	−	0.772421i	$-0.280954\pi$
$197$	17.8284	1.27022	0.635111	+	0.772421i	$0.280954\pi$
$198$	0	0
$199$	−22.9706	−1.62834	−0.814170	−	0.580627i	$-0.802808\pi$
$199$	−22.9706	−1.62834	−0.814170	+	0.580627i	$0.802808\pi$
$200$	0	0
$201$	−3.65685	−0.257935
$202$	0	0
$203$	12.1421	0.852211
$204$	0	0
$205$	10.8284	0.756290
$206$	0	0
$207$	6.34315	0.440879
$208$	0	0
$209$	−19.3137	−1.33596
$210$	0	0
$211$	−1.34315	−0.0924660	−0.0462330	−	0.998931i	$-0.514722\pi$
$211$	−1.34315	−0.0924660	−0.0462330	+	0.998931i	$0.514722\pi$
$212$	0	0
$213$	10.1716	0.696945
$214$	0	0
$215$	−11.4853	−0.783290
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5.17157	−0.349463
$220$	0	0
$221$	−6.65685	−0.447788
$222$	0	0
$223$	−10.1716	−0.681139	−0.340569	−	0.940219i	$-0.610620\pi$
$223$	−10.1716	−0.681139	−0.340569	+	0.940219i	$0.610620\pi$
$224$	0	0
$225$	−19.3137	−1.28758
$226$	0	0
$227$	9.17157	0.608739	0.304369	−	0.952554i	$-0.401554\pi$
$227$	9.17157	0.608739	0.304369	+	0.952554i	$0.401554\pi$
$228$	0	0
$229$	−10.1716	−0.672156	−0.336078	−	0.941834i	$-0.609101\pi$
$229$	−10.1716	−0.672156	−0.336078	+	0.941834i	$0.609101\pi$
$230$	0	0
$231$	−18.4853	−1.21624
$232$	0	0
$233$	20.3137	1.33080	0.665398	−	0.746489i	$-0.268262\pi$
$233$	20.3137	1.33080	0.665398	+	0.746489i	$0.268262\pi$
$234$	0	0
$235$	−43.9706	−2.86832
$236$	0	0
$237$	7.65685	0.497366
$238$	0	0
$239$	19.1421	1.23820	0.619101	−	0.785311i	$-0.287497\pi$
$239$	19.1421	1.23820	0.619101	+	0.785311i	$0.287497\pi$
$240$	0	0
$241$	3.31371	0.213455	0.106727	−	0.994288i	$-0.465963\pi$
$241$	3.31371	0.213455	0.106727	+	0.994288i	$0.465963\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	29.3137	1.87278
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	26.6274	1.68071	0.840354	−	0.542038i	$-0.182347\pi$
$251$	26.6274	1.68071	0.840354	+	0.542038i	$0.182347\pi$
$252$	0	0
$253$	−15.3137	−0.962765
$254$	0	0
$255$	−25.4853	−1.59595
$256$	0	0
$257$	−1.34315	−0.0837831	−0.0418916	−	0.999122i	$-0.513338\pi$
$257$	−1.34315	−0.0837831	−0.0418916	+	0.999122i	$0.513338\pi$
$258$	0	0
$259$	14.6569	0.910733
$260$	0	0
$261$	−6.34315	−0.392631
$262$	0	0
$263$	26.1421	1.61199	0.805997	−	0.591920i	$-0.201630\pi$
$263$	26.1421	1.61199	0.805997	+	0.591920i	$0.201630\pi$
$264$	0	0
$265$	12.1421	0.745885
$266$	0	0
$267$	17.6569	1.08058
$268$	0	0
$269$	−19.7990	−1.20717	−0.603583	−	0.797300i	$-0.706261\pi$
$269$	−19.7990	−1.20717	−0.603583	+	0.797300i	$0.706261\pi$
$270$	0	0
$271$	−2.51472	−0.152758	−0.0763791	−	0.997079i	$-0.524336\pi$
$271$	−2.51472	−0.152758	−0.0763791	+	0.997079i	$0.524336\pi$
$272$	0	0
$273$	3.82843	0.231707
$274$	0	0
$275$	46.6274	2.81174
$276$	0	0
$277$	−4.48528	−0.269494	−0.134747	−	0.990880i	$-0.543022\pi$
$277$	−4.48528	−0.269494	−0.134747	+	0.990880i	$0.543022\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.97056	−0.296519	−0.148259	−	0.988948i	$-0.547367\pi$
$281$	−4.97056	−0.296519	−0.148259	+	0.988948i	$0.547367\pi$
$282$	0	0
$283$	−14.6274	−0.869510	−0.434755	−	0.900549i	$-0.643165\pi$
$283$	−14.6274	−0.869510	−0.434755	+	0.900549i	$0.643165\pi$
$284$	0	0
$285$	15.3137	0.907106
$286$	0	0
$287$	10.8284	0.639182
$288$	0	0
$289$	27.3137	1.60669
$290$	0	0
$291$	14.4853	0.849142
$292$	0	0
$293$	12.7990	0.747725	0.373862	−	0.927484i	$-0.378033\pi$
$293$	12.7990	0.747725	0.373862	+	0.927484i	$0.378033\pi$
$294$	0	0
$295$	−19.7990	−1.15274
$296$	0	0
$297$	24.1421	1.40087
$298$	0	0
$299$	3.17157	0.183417
$300$	0	0
$301$	−11.4853	−0.662001
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41.4558	2.37375
$306$	0	0
$307$	13.1716	0.751741	0.375871	−	0.926672i	$-0.377344\pi$
$307$	13.1716	0.751741	0.375871	+	0.926672i	$0.377344\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−24.2843	−1.37703	−0.688517	−	0.725220i	$-0.741738\pi$
$311$	−24.2843	−1.37703	−0.688517	+	0.725220i	$0.741738\pi$
$312$	0	0
$313$	19.9706	1.12880	0.564401	−	0.825500i	$-0.309107\pi$
$313$	19.9706	1.12880	0.564401	+	0.825500i	$0.309107\pi$
$314$	0	0
$315$	−29.3137	−1.65164
$316$	0	0
$317$	29.3137	1.64642	0.823211	−	0.567736i	$-0.192180\pi$
$317$	29.3137	1.64642	0.823211	+	0.567736i	$0.192180\pi$
$318$	0	0
$319$	15.3137	0.857403
$320$	0	0
$321$	15.3137	0.854728
$322$	0	0
$323$	−26.6274	−1.48159
$324$	0	0
$325$	−9.65685	−0.535666
$326$	0	0
$327$	−3.82843	−0.211713
$328$	0	0
$329$	−43.9706	−2.42418
$330$	0	0
$331$	10.9706	0.602997	0.301498	−	0.953467i	$-0.402513\pi$
$331$	10.9706	0.602997	0.301498	+	0.953467i	$0.402513\pi$
$332$	0	0
$333$	−7.65685	−0.419593
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	−1.97056	−0.107343	−0.0536717	−	0.998559i	$-0.517092\pi$
$337$	−1.97056	−0.107343	−0.0536717	+	0.998559i	$0.517092\pi$
$338$	0	0
$339$	3.65685	0.198613
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.51472	0.135782
$344$	0	0
$345$	12.1421	0.653711
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	25.4853	1.36420	0.682098	−	0.731261i	$-0.261068\pi$
$349$	25.4853	1.36420	0.682098	+	0.731261i	$0.261068\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−17.3137	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$353$	−17.3137	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$354$	0	0
$355$	−38.9411	−2.06678
$356$	0	0
$357$	−25.4853	−1.34882
$358$	0	0
$359$	−21.6569	−1.14301	−0.571503	−	0.820600i	$-0.693639\pi$
$359$	−21.6569	−1.14301	−0.571503	+	0.820600i	$0.693639\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−12.3137	−0.646302
$364$	0	0
$365$	19.7990	1.03633
$366$	0	0
$367$	31.9411	1.66731	0.833657	−	0.552283i	$-0.186243\pi$
$367$	31.9411	1.66731	0.833657	+	0.552283i	$0.186243\pi$
$368$	0	0
$369$	−5.65685	−0.294484
$370$	0	0
$371$	12.1421	0.630388
$372$	0	0
$373$	−8.97056	−0.464478	−0.232239	−	0.972659i	$-0.574605\pi$
$373$	−8.97056	−0.464478	−0.232239	+	0.972659i	$0.574605\pi$
$374$	0	0
$375$	−17.8284	−0.920656
$376$	0	0
$377$	−3.17157	−0.163344
$378$	0	0
$379$	−23.4558	−1.20485	−0.602423	−	0.798177i	$-0.705798\pi$
$379$	−23.4558	−1.20485	−0.602423	+	0.798177i	$0.705798\pi$
$380$	0	0
$381$	−18.4853	−0.947030
$382$	0	0
$383$	17.8284	0.910990	0.455495	−	0.890238i	$-0.349462\pi$
$383$	17.8284	0.910990	0.455495	+	0.890238i	$0.349462\pi$
$384$	0	0
$385$	70.7696	3.60675
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	26.1421	1.32546	0.662729	−	0.748859i	$-0.269398\pi$
$389$	26.1421	1.32546	0.662729	+	0.748859i	$0.269398\pi$
$390$	0	0
$391$	−21.1127	−1.06772
$392$	0	0
$393$	−2.65685	−0.134021
$394$	0	0
$395$	−29.3137	−1.47493
$396$	0	0
$397$	1.31371	0.0659331	0.0329666	−	0.999456i	$-0.489505\pi$
$397$	1.31371	0.0659331	0.0329666	+	0.999456i	$0.489505\pi$
$398$	0	0
$399$	15.3137	0.766644
$400$	0	0
$401$	−11.1716	−0.557882	−0.278941	−	0.960308i	$-0.589983\pi$
$401$	−11.1716	−0.557882	−0.278941	+	0.960308i	$0.589983\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.82843	0.190236
$406$	0	0
$407$	18.4853	0.916281
$408$	0	0
$409$	−1.51472	−0.0748980	−0.0374490	−	0.999299i	$-0.511923\pi$
$409$	−1.51472	−0.0748980	−0.0374490	+	0.999299i	$0.511923\pi$
$410$	0	0
$411$	20.1421	0.993538
$412$	0	0
$413$	−19.7990	−0.974245
$414$	0	0
$415$	22.9706	1.12758
$416$	0	0
$417$	−18.6569	−0.913630
$418$	0	0
$419$	−19.3431	−0.944975	−0.472487	−	0.881338i	$-0.656644\pi$
$419$	−19.3431	−0.944975	−0.472487	+	0.881338i	$0.656644\pi$
$420$	0	0
$421$	−8.85786	−0.431706	−0.215853	−	0.976426i	$-0.569253\pi$
$421$	−8.85786	−0.431706	−0.215853	+	0.976426i	$0.569253\pi$
$422$	0	0
$423$	22.9706	1.11687
$424$	0	0
$425$	64.2843	3.11825
$426$	0	0
$427$	41.4558	2.00619
$428$	0	0
$429$	4.82843	0.233119
$430$	0	0
$431$	−17.8284	−0.858765	−0.429383	−	0.903123i	$-0.641269\pi$
$431$	−17.8284	−0.858765	−0.429383	+	0.903123i	$0.641269\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	−12.1421	−0.582171
$436$	0	0
$437$	12.6863	0.606868
$438$	0	0
$439$	−4.48528	−0.214071	−0.107035	−	0.994255i	$-0.534136\pi$
$439$	−4.48528	−0.214071	−0.107035	+	0.994255i	$0.534136\pi$
$440$	0	0
$441$	−15.3137	−0.729224
$442$	0	0
$443$	−0.313708	−0.0149047	−0.00745237	−	0.999972i	$-0.502372\pi$
$443$	−0.313708	−0.0149047	−0.00745237	+	0.999972i	$0.502372\pi$
$444$	0	0
$445$	−67.5980	−3.20445
$446$	0	0
$447$	1.31371	0.0621363
$448$	0	0
$449$	12.1421	0.573023	0.286511	−	0.958077i	$-0.407504\pi$
$449$	12.1421	0.573023	0.286511	+	0.958077i	$0.407504\pi$
$450$	0	0
$451$	13.6569	0.643076
$452$	0	0
$453$	−2.51472	−0.118152
$454$	0	0
$455$	−14.6569	−0.687124
$456$	0	0
$457$	−20.9706	−0.980962	−0.490481	−	0.871452i	$-0.663179\pi$
$457$	−20.9706	−0.980962	−0.490481	+	0.871452i	$0.663179\pi$
$458$	0	0
$459$	33.2843	1.55358
$460$	0	0
$461$	−11.4853	−0.534923	−0.267461	−	0.963569i	$-0.586185\pi$
$461$	−11.4853	−0.534923	−0.267461	+	0.963569i	$0.586185\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.3431	−1.40411	−0.702057	−	0.712121i	$-0.747735\pi$
$467$	−30.3431	−1.40411	−0.702057	+	0.712121i	$0.747735\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	4.48528	0.206671
$472$	0	0
$473$	−14.4853	−0.666034
$474$	0	0
$475$	−38.6274	−1.77235
$476$	0	0
$477$	−6.34315	−0.290433
$478$	0	0
$479$	19.1421	0.874627	0.437313	−	0.899309i	$-0.355930\pi$
$479$	19.1421	0.874627	0.437313	+	0.899309i	$0.355930\pi$
$480$	0	0
$481$	−3.82843	−0.174561
$482$	0	0
$483$	12.1421	0.552486
$484$	0	0
$485$	−55.4558	−2.51812
$486$	0	0
$487$	−8.97056	−0.406495	−0.203247	−	0.979127i	$-0.565150\pi$
$487$	−8.97056	−0.406495	−0.203247	+	0.979127i	$0.565150\pi$
$488$	0	0
$489$	12.8284	0.580122
$490$	0	0
$491$	17.6863	0.798171	0.399086	−	0.916914i	$-0.369328\pi$
$491$	17.6863	0.798171	0.399086	+	0.916914i	$0.369328\pi$
$492$	0	0
$493$	21.1127	0.950868
$494$	0	0
$495$	−36.9706	−1.66170
$496$	0	0
$497$	−38.9411	−1.74675
$498$	0	0
$499$	−18.1421	−0.812154	−0.406077	−	0.913839i	$-0.633103\pi$
$499$	−18.1421	−0.812154	−0.406077	+	0.913839i	$0.633103\pi$
$500$	0	0
$501$	−21.6569	−0.967557
$502$	0	0
$503$	15.3137	0.682805	0.341402	−	0.939917i	$-0.389098\pi$
$503$	15.3137	0.682805	0.341402	+	0.939917i	$0.389098\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	38.2843	1.69692	0.848460	−	0.529259i	$-0.177530\pi$
$509$	38.2843	1.69692	0.848460	+	0.529259i	$0.177530\pi$
$510$	0	0
$511$	19.7990	0.875856
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	−53.5980	−2.36181
$516$	0	0
$517$	−55.4558	−2.43895
$518$	0	0
$519$	22.9706	1.00830
$520$	0	0
$521$	14.3137	0.627095	0.313547	−	0.949573i	$-0.398483\pi$
$521$	14.3137	0.627095	0.313547	+	0.949573i	$0.398483\pi$
$522$	0	0
$523$	9.65685	0.422265	0.211132	−	0.977457i	$-0.432285\pi$
$523$	9.65685	0.422265	0.211132	+	0.977457i	$0.432285\pi$
$524$	0	0
$525$	−36.9706	−1.61353
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	10.3431	0.448854
$532$	0	0
$533$	−2.82843	−0.122513
$534$	0	0
$535$	−58.6274	−2.53468
$536$	0	0
$537$	−8.31371	−0.358763
$538$	0	0
$539$	36.9706	1.59243
$540$	0	0
$541$	−16.5147	−0.710023	−0.355012	−	0.934862i	$-0.615523\pi$
$541$	−16.5147	−0.710023	−0.355012	+	0.934862i	$0.615523\pi$
$542$	0	0
$543$	−22.9706	−0.985761
$544$	0	0
$545$	14.6569	0.627831
$546$	0	0
$547$	5.62742	0.240611	0.120305	−	0.992737i	$-0.461613\pi$
$547$	5.62742	0.240611	0.120305	+	0.992737i	$0.461613\pi$
$548$	0	0
$549$	−21.6569	−0.924292
$550$	0	0
$551$	−12.6863	−0.540454
$552$	0	0
$553$	−29.3137	−1.24655
$554$	0	0
$555$	−14.6569	−0.622149
$556$	0	0
$557$	5.14214	0.217879	0.108940	−	0.994048i	$-0.465254\pi$
$557$	5.14214	0.217879	0.108940	+	0.994048i	$0.465254\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	−32.1421	−1.35704
$562$	0	0
$563$	−38.5980	−1.62671	−0.813355	−	0.581767i	$-0.802362\pi$
$563$	−38.5980	−1.62671	−0.813355	+	0.581767i	$0.802362\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	0	0
$567$	3.82843	0.160779
$568$	0	0
$569$	−1.97056	−0.0826103	−0.0413051	−	0.999147i	$-0.513152\pi$
$569$	−1.97056	−0.0826103	−0.0413051	+	0.999147i	$0.513152\pi$
$570$	0	0
$571$	−22.6569	−0.948160	−0.474080	−	0.880482i	$-0.657219\pi$
$571$	−22.6569	−0.948160	−0.474080	+	0.880482i	$0.657219\pi$
$572$	0	0
$573$	−3.17157	−0.132494
$574$	0	0
$575$	−30.6274	−1.27725
$576$	0	0
$577$	33.4558	1.39279	0.696393	−	0.717661i	$-0.254787\pi$
$577$	33.4558	1.39279	0.696393	+	0.717661i	$0.254787\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	22.9706	0.952980
$582$	0	0
$583$	15.3137	0.634229
$584$	0	0
$585$	7.65685	0.316572
$586$	0	0
$587$	−25.1127	−1.03651	−0.518256	−	0.855226i	$-0.673419\pi$
$587$	−25.1127	−1.03651	−0.518256	+	0.855226i	$0.673419\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−17.8284	−0.733363
$592$	0	0
$593$	−8.14214	−0.334357	−0.167179	−	0.985927i	$-0.553466\pi$
$593$	−8.14214	−0.334357	−0.167179	+	0.985927i	$0.553466\pi$
$594$	0	0
$595$	97.5685	3.99992
$596$	0	0
$597$	22.9706	0.940123
$598$	0	0
$599$	−1.85786	−0.0759103	−0.0379551	−	0.999279i	$-0.512084\pi$
$599$	−1.85786	−0.0759103	−0.0379551	+	0.999279i	$0.512084\pi$
$600$	0	0
$601$	1.68629	0.0687853	0.0343926	−	0.999408i	$-0.489050\pi$
$601$	1.68629	0.0687853	0.0343926	+	0.999408i	$0.489050\pi$
$602$	0	0
$603$	−7.31371	−0.297837
$604$	0	0
$605$	47.1421	1.91660
$606$	0	0
$607$	−10.8284	−0.439512	−0.219756	−	0.975555i	$-0.570526\pi$
$607$	−10.8284	−0.439512	−0.219756	+	0.975555i	$0.570526\pi$
$608$	0	0
$609$	−12.1421	−0.492024
$610$	0	0
$611$	11.4853	0.464645
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	0	0
$615$	−10.8284	−0.436644
$616$	0	0
$617$	−0.686292	−0.0276291	−0.0138145	−	0.999905i	$-0.504397\pi$
$617$	−0.686292	−0.0276291	−0.0138145	+	0.999905i	$0.504397\pi$
$618$	0	0
$619$	18.1421	0.729194	0.364597	−	0.931165i	$-0.381207\pi$
$619$	18.1421	0.729194	0.364597	+	0.931165i	$0.381207\pi$
$620$	0	0
$621$	−15.8579	−0.636354
$622$	0	0
$623$	−67.5980	−2.70826
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	19.3137	0.771315
$628$	0	0
$629$	25.4853	1.01616
$630$	0	0
$631$	−12.7990	−0.509520	−0.254760	−	0.967004i	$-0.581996\pi$
$631$	−12.7990	−0.509520	−0.254760	+	0.967004i	$0.581996\pi$
$632$	0	0
$633$	1.34315	0.0533853
$634$	0	0
$635$	70.7696	2.80840
$636$	0	0
$637$	−7.65685	−0.303376
$638$	0	0
$639$	20.3431	0.804762
$640$	0	0
$641$	−22.9706	−0.907283	−0.453641	−	0.891184i	$-0.649875\pi$
$641$	−22.9706	−0.907283	−0.453641	+	0.891184i	$0.649875\pi$
$642$	0	0
$643$	−48.6274	−1.91768	−0.958839	−	0.283950i	$-0.908355\pi$
$643$	−48.6274	−1.91768	−0.958839	+	0.283950i	$0.908355\pi$
$644$	0	0
$645$	11.4853	0.452233
$646$	0	0
$647$	17.1716	0.675084	0.337542	−	0.941310i	$-0.390404\pi$
$647$	17.1716	0.675084	0.337542	+	0.941310i	$0.390404\pi$
$648$	0	0
$649$	−24.9706	−0.980180
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.3137	0.599272	0.299636	−	0.954054i	$-0.403135\pi$
$653$	15.3137	0.599272	0.299636	+	0.954054i	$0.403135\pi$
$654$	0	0
$655$	10.1716	0.397436
$656$	0	0
$657$	−10.3431	−0.403525
$658$	0	0
$659$	−4.97056	−0.193626	−0.0968128	−	0.995303i	$-0.530865\pi$
$659$	−4.97056	−0.193626	−0.0968128	+	0.995303i	$0.530865\pi$
$660$	0	0
$661$	1.31371	0.0510973	0.0255487	−	0.999674i	$-0.491867\pi$
$661$	1.31371	0.0510973	0.0255487	+	0.999674i	$0.491867\pi$
$662$	0	0
$663$	6.65685	0.258531
$664$	0	0
$665$	−58.6274	−2.27347
$666$	0	0
$667$	−10.0589	−0.389481
$668$	0	0
$669$	10.1716	0.393256
$670$	0	0
$671$	52.2843	2.01841
$672$	0	0
$673$	12.3137	0.474659	0.237329	−	0.971429i	$-0.423728\pi$
$673$	12.3137	0.474659	0.237329	+	0.971429i	$0.423728\pi$
$674$	0	0
$675$	48.2843	1.85846
$676$	0	0
$677$	−29.3137	−1.12662	−0.563309	−	0.826247i	$-0.690472\pi$
$677$	−29.3137	−1.12662	−0.563309	+	0.826247i	$0.690472\pi$
$678$	0	0
$679$	−55.4558	−2.12820
$680$	0	0
$681$	−9.17157	−0.351455
$682$	0	0
$683$	−39.6569	−1.51743	−0.758714	−	0.651424i	$-0.774172\pi$
$683$	−39.6569	−1.51743	−0.758714	+	0.651424i	$0.774172\pi$
$684$	0	0
$685$	−77.1127	−2.94632
$686$	0	0
$687$	10.1716	0.388070
$688$	0	0
$689$	−3.17157	−0.120827
$690$	0	0
$691$	−7.02944	−0.267412	−0.133706	−	0.991021i	$-0.542688\pi$
$691$	−7.02944	−0.267412	−0.133706	+	0.991021i	$0.542688\pi$
$692$	0	0
$693$	−36.9706	−1.40440
$694$	0	0
$695$	71.4264	2.70936
$696$	0	0
$697$	18.8284	0.713178
$698$	0	0
$699$	−20.3137	−0.768335
$700$	0	0
$701$	−12.1421	−0.458602	−0.229301	−	0.973356i	$-0.573644\pi$
$701$	−12.1421	−0.458602	−0.229301	+	0.973356i	$0.573644\pi$
$702$	0	0
$703$	−15.3137	−0.577567
$704$	0	0
$705$	43.9706	1.65603
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.65685	0.287559	0.143780	−	0.989610i	$-0.454074\pi$
$709$	7.65685	0.287559	0.143780	+	0.989610i	$0.454074\pi$
$710$	0	0
$711$	15.3137	0.574309
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−18.4853	−0.691310
$716$	0	0
$717$	−19.1421	−0.714876
$718$	0	0
$719$	−22.9706	−0.856657	−0.428329	−	0.903623i	$-0.640897\pi$
$719$	−22.9706	−0.856657	−0.428329	+	0.903623i	$0.640897\pi$
$720$	0	0
$721$	−53.5980	−1.99609
$722$	0	0
$723$	−3.31371	−0.123238
$724$	0	0
$725$	30.6274	1.13747
$726$	0	0
$727$	−18.4853	−0.685581	−0.342791	−	0.939412i	$-0.611372\pi$
$727$	−18.4853	−0.685581	−0.342791	+	0.939412i	$0.611372\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−19.9706	−0.738638
$732$	0	0
$733$	35.7696	1.32118	0.660589	−	0.750747i	$-0.270306\pi$
$733$	35.7696	1.32118	0.660589	+	0.750747i	$0.270306\pi$
$734$	0	0
$735$	−29.3137	−1.08125
$736$	0	0
$737$	17.6569	0.650399
$738$	0	0
$739$	31.5147	1.15929	0.579644	−	0.814870i	$-0.303192\pi$
$739$	31.5147	1.15929	0.579644	+	0.814870i	$0.303192\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	40.7990	1.49677	0.748385	−	0.663265i	$-0.230830\pi$
$743$	40.7990	1.49677	0.748385	+	0.663265i	$0.230830\pi$
$744$	0	0
$745$	−5.02944	−0.184264
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−58.6274	−2.14220
$750$	0	0
$751$	−23.5147	−0.858064	−0.429032	−	0.903289i	$-0.641145\pi$
$751$	−23.5147	−0.858064	−0.429032	+	0.903289i	$0.641145\pi$
$752$	0	0
$753$	−26.6274	−0.970357
$754$	0	0
$755$	9.62742	0.350378
$756$	0	0
$757$	31.1716	1.13295	0.566475	−	0.824079i	$-0.308307\pi$
$757$	31.1716	1.13295	0.566475	+	0.824079i	$0.308307\pi$
$758$	0	0
$759$	15.3137	0.555852
$760$	0	0
$761$	−33.1716	−1.20247	−0.601234	−	0.799073i	$-0.705324\pi$
$761$	−33.1716	−1.20247	−0.601234	+	0.799073i	$0.705324\pi$
$762$	0	0
$763$	14.6569	0.530614
$764$	0	0
$765$	−50.9706	−1.84284
$766$	0	0
$767$	5.17157	0.186735
$768$	0	0
$769$	−13.6569	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$769$	−13.6569	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$770$	0	0
$771$	1.34315	0.0483722
$772$	0	0
$773$	−26.7990	−0.963893	−0.481946	−	0.876201i	$-0.660070\pi$
$773$	−26.7990	−0.963893	−0.481946	+	0.876201i	$0.660070\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−14.6569	−0.525812
$778$	0	0
$779$	−11.3137	−0.405356
$780$	0	0
$781$	−49.1127	−1.75739
$782$	0	0
$783$	15.8579	0.566714
$784$	0	0
$785$	−17.1716	−0.612880
$786$	0	0
$787$	−11.5147	−0.410455	−0.205228	−	0.978714i	$-0.565793\pi$
$787$	−11.5147	−0.410455	−0.205228	+	0.978714i	$0.565793\pi$
$788$	0	0
$789$	−26.1421	−0.930685
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−10.8284	−0.384529
$794$	0	0
$795$	−12.1421	−0.430637
$796$	0	0
$797$	34.3431	1.21650	0.608248	−	0.793747i	$-0.291872\pi$
$797$	34.3431	1.21650	0.608248	+	0.793747i	$0.291872\pi$
$798$	0	0
$799$	−76.4558	−2.70481
$800$	0	0
$801$	35.3137	1.24775
$802$	0	0
$803$	24.9706	0.881192
$804$	0	0
$805$	−46.4853	−1.63839
$806$	0	0
$807$	19.7990	0.696957
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	28.9706	1.01729	0.508647	−	0.860975i	$-0.330146\pi$
$811$	28.9706	1.01729	0.508647	+	0.860975i	$0.330146\pi$
$812$	0	0
$813$	2.51472	0.0881950
$814$	0	0
$815$	−49.1127	−1.72034
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	7.65685	0.267552
$820$	0	0
$821$	−11.4853	−0.400839	−0.200420	−	0.979710i	$-0.564231\pi$
$821$	−11.4853	−0.400839	−0.200420	+	0.979710i	$0.564231\pi$
$822$	0	0
$823$	28.7696	1.00284	0.501422	−	0.865203i	$-0.332811\pi$
$823$	28.7696	1.00284	0.501422	+	0.865203i	$0.332811\pi$
$824$	0	0
$825$	−46.6274	−1.62336
$826$	0	0
$827$	30.6274	1.06502	0.532510	−	0.846424i	$-0.321249\pi$
$827$	30.6274	1.06502	0.532510	+	0.846424i	$0.321249\pi$
$828$	0	0
$829$	−22.9706	−0.797801	−0.398900	−	0.916994i	$-0.630608\pi$
$829$	−22.9706	−0.797801	−0.398900	+	0.916994i	$0.630608\pi$
$830$	0	0
$831$	4.48528	0.155593
$832$	0	0
$833$	50.9706	1.76603
$834$	0	0
$835$	82.9117	2.86928
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.6863	−0.437979	−0.218990	−	0.975727i	$-0.570276\pi$
$839$	−12.6863	−0.437979	−0.218990	+	0.975727i	$0.570276\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	0	0
$843$	4.97056	0.171195
$844$	0	0
$845$	3.82843	0.131702
$846$	0	0
$847$	47.1421	1.61982
$848$	0	0
$849$	14.6274	0.502012
$850$	0	0
$851$	−12.1421	−0.416227
$852$	0	0
$853$	35.7696	1.22473	0.612363	−	0.790577i	$-0.290219\pi$
$853$	35.7696	1.22473	0.612363	+	0.790577i	$0.290219\pi$
$854$	0	0
$855$	30.6274	1.04744
$856$	0	0
$857$	41.3137	1.41125	0.705625	−	0.708586i	$-0.250666\pi$
$857$	41.3137	1.41125	0.705625	+	0.708586i	$0.250666\pi$
$858$	0	0
$859$	−4.97056	−0.169593	−0.0847967	−	0.996398i	$-0.527024\pi$
$859$	−4.97056	−0.169593	−0.0847967	+	0.996398i	$0.527024\pi$
$860$	0	0
$861$	−10.8284	−0.369032
$862$	0	0
$863$	−14.1127	−0.480402	−0.240201	−	0.970723i	$-0.577213\pi$
$863$	−14.1127	−0.480402	−0.240201	+	0.970723i	$0.577213\pi$
$864$	0	0
$865$	−87.9411	−2.99009
$866$	0	0
$867$	−27.3137	−0.927622
$868$	0	0
$869$	−36.9706	−1.25414
$870$	0	0
$871$	−3.65685	−0.123908
$872$	0	0
$873$	28.9706	0.980505
$874$	0	0
$875$	68.2548	2.30743
$876$	0	0
$877$	−16.5147	−0.557662	−0.278831	−	0.960340i	$-0.589947\pi$
$877$	−16.5147	−0.557662	−0.278831	+	0.960340i	$0.589947\pi$
$878$	0	0
$879$	−12.7990	−0.431699
$880$	0	0
$881$	40.3137	1.35820	0.679102	−	0.734044i	$-0.262370\pi$
$881$	40.3137	1.35820	0.679102	+	0.734044i	$0.262370\pi$
$882$	0	0
$883$	−1.97056	−0.0663147	−0.0331574	−	0.999450i	$-0.510556\pi$
$883$	−1.97056	−0.0663147	−0.0331574	+	0.999450i	$0.510556\pi$
$884$	0	0
$885$	19.7990	0.665536
$886$	0	0
$887$	−26.1421	−0.877767	−0.438884	−	0.898544i	$-0.644626\pi$
$887$	−26.1421	−0.877767	−0.438884	+	0.898544i	$0.644626\pi$
$888$	0	0
$889$	70.7696	2.37353
$890$	0	0
$891$	4.82843	0.161758
$892$	0	0
$893$	45.9411	1.53736
$894$	0	0
$895$	31.8284	1.06391
$896$	0	0
$897$	−3.17157	−0.105896
$898$	0	0
$899$	0	0
$900$	0	0
$901$	21.1127	0.703366
$902$	0	0
$903$	11.4853	0.382206
$904$	0	0
$905$	87.9411	2.92326
$906$	0	0
$907$	−24.9411	−0.828156	−0.414078	−	0.910241i	$-0.635896\pi$
$907$	−24.9411	−0.828156	−0.414078	+	0.910241i	$0.635896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.4558	1.83733	0.918667	−	0.395033i	$-0.129267\pi$
$911$	55.4558	1.83733	0.918667	+	0.395033i	$0.129267\pi$
$912$	0	0
$913$	28.9706	0.958786
$914$	0	0
$915$	−41.4558	−1.37049
$916$	0	0
$917$	10.1716	0.335895
$918$	0	0
$919$	15.3137	0.505153	0.252576	−	0.967577i	$-0.418722\pi$
$919$	15.3137	0.505153	0.252576	+	0.967577i	$0.418722\pi$
$920$	0	0
$921$	−13.1716	−0.434018
$922$	0	0
$923$	10.1716	0.334801
$924$	0	0
$925$	36.9706	1.21558
$926$	0	0
$927$	28.0000	0.919641
$928$	0	0
$929$	−28.6863	−0.941167	−0.470583	−	0.882356i	$-0.655956\pi$
$929$	−28.6863	−0.941167	−0.470583	+	0.882356i	$0.655956\pi$
$930$	0	0
$931$	−30.6274	−1.00377
$932$	0	0
$933$	24.2843	0.795031
$934$	0	0
$935$	123.054	4.02429
$936$	0	0
$937$	−15.6569	−0.511487	−0.255744	−	0.966745i	$-0.582320\pi$
$937$	−15.6569	−0.511487	−0.255744	+	0.966745i	$0.582320\pi$
$938$	0	0
$939$	−19.9706	−0.651715
$940$	0	0
$941$	−40.7990	−1.33001	−0.665005	−	0.746839i	$-0.731570\pi$
$941$	−40.7990	−1.33001	−0.665005	+	0.746839i	$0.731570\pi$
$942$	0	0
$943$	−8.97056	−0.292122
$944$	0	0
$945$	73.2843	2.38394
$946$	0	0
$947$	−37.1127	−1.20600	−0.603000	−	0.797741i	$-0.706028\pi$
$947$	−37.1127	−1.20600	−0.603000	+	0.797741i	$0.706028\pi$
$948$	0	0
$949$	−5.17157	−0.167876
$950$	0	0
$951$	−29.3137	−0.950562
$952$	0	0
$953$	−5.68629	−0.184197	−0.0920985	−	0.995750i	$-0.529357\pi$
$953$	−5.68629	−0.184197	−0.0920985	+	0.995750i	$0.529357\pi$
$954$	0	0
$955$	12.1421	0.392910
$956$	0	0
$957$	−15.3137	−0.495022
$958$	0	0
$959$	−77.1127	−2.49010
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	30.6274	0.986955
$964$	0	0
$965$	−22.9706	−0.739449
$966$	0	0
$967$	15.2010	0.488832	0.244416	−	0.969671i	$-0.421404\pi$
$967$	15.2010	0.488832	0.244416	+	0.969671i	$0.421404\pi$
$968$	0	0
$969$	26.6274	0.855396
$970$	0	0
$971$	−36.5980	−1.17449	−0.587243	−	0.809411i	$-0.699787\pi$
$971$	−36.5980	−1.17449	−0.587243	+	0.809411i	$0.699787\pi$
$972$	0	0
$973$	71.4264	2.28983
$974$	0	0
$975$	9.65685	0.309267
$976$	0	0
$977$	2.28427	0.0730803	0.0365402	−	0.999332i	$-0.488366\pi$
$977$	2.28427	0.0730803	0.0365402	+	0.999332i	$0.488366\pi$
$978$	0	0
$979$	−85.2548	−2.72476
$980$	0	0
$981$	−7.65685	−0.244465
$982$	0	0
$983$	21.7696	0.694341	0.347170	−	0.937802i	$-0.387143\pi$
$983$	21.7696	0.694341	0.347170	+	0.937802i	$0.387143\pi$
$984$	0	0
$985$	68.2548	2.17478
$986$	0	0
$987$	43.9706	1.39960
$988$	0	0
$989$	9.51472	0.302550
$990$	0	0
$991$	19.0294	0.604490	0.302245	−	0.953230i	$-0.402264\pi$
$991$	19.0294	0.604490	0.302245	+	0.953230i	$0.402264\pi$
$992$	0	0
$993$	−10.9706	−0.348140
$994$	0	0
$995$	−87.9411	−2.78792
$996$	0	0
$997$	27.4558	0.869535	0.434768	−	0.900543i	$-0.356830\pi$
$997$	27.4558	0.869535	0.434768	+	0.900543i	$0.356830\pi$
$998$	0	0
$999$	19.1421	0.605630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.r.1.2		2
4.3	odd	2		3328.2.a.ba.1.2		2
8.3	odd	2		3328.2.a.o.1.1		2
8.5	even	2		3328.2.a.z.1.1		2
16.3	odd	4		1664.2.b.i.833.3	yes	4
16.5	even	4		1664.2.b.e.833.4	yes	4
16.11	odd	4		1664.2.b.i.833.2	yes	4
16.13	even	4		1664.2.b.e.833.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.e.833.1	✓	4	16.13	even	4
1664.2.b.e.833.4	yes	4	16.5	even	4
1664.2.b.i.833.2	yes	4	16.11	odd	4
1664.2.b.i.833.3	yes	4	16.3	odd	4
3328.2.a.o.1.1		2	8.3	odd	2
3328.2.a.r.1.2		2	1.1	even	1	trivial
3328.2.a.z.1.1		2	8.5	even	2
3328.2.a.ba.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.82843 q^{5} +3.82843 q^{7} -2.00000 q^{9} +4.82843 q^{11} -1.00000 q^{13} -3.82843 q^{15} +6.65685 q^{17} -4.00000 q^{19} -3.82843 q^{21} -3.17157 q^{23} +9.65685 q^{25} +5.00000 q^{27} +3.17157 q^{29} -4.82843 q^{33} +14.6569 q^{35} +3.82843 q^{37} +1.00000 q^{39} +2.82843 q^{41} -3.00000 q^{43} -7.65685 q^{45} -11.4853 q^{47} +7.65685 q^{49} -6.65685 q^{51} +3.17157 q^{53} +18.4853 q^{55} +4.00000 q^{57} -5.17157 q^{59} +10.8284 q^{61} -7.65685 q^{63} -3.82843 q^{65} +3.65685 q^{67} +3.17157 q^{69} -10.1716 q^{71} +5.17157 q^{73} -9.65685 q^{75} +18.4853 q^{77} -7.65685 q^{79} +1.00000 q^{81} +6.00000 q^{83} +25.4853 q^{85} -3.17157 q^{87} -17.6569 q^{89} -3.82843 q^{91} -15.3137 q^{95} -14.4853 q^{97} -9.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} - 2 q^{21} - 12 q^{23} + 8 q^{25} + 10 q^{27} + 12 q^{29} - 4 q^{33} + 18 q^{35} + 2 q^{37} + 2 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 - 4 * q^9 + 4 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 - 2 * q^21 - 12 * q^23 + 8 * q^25 + 10 * q^27 + 12 * q^29 - 4 * q^33 + 18 * q^35 + 2 * q^37 + 2 * q^39 - 6 * q^43 - 4 * q^45 - 6 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 + 20 * q^55 + 8 * q^57 - 16 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 - 4 * q^67 + 12 * q^69 - 26 * q^71 + 16 * q^73 - 8 * q^75 + 20 * q^77 - 4 * q^79 + 2 * q^81 + 12 * q^83 + 34 * q^85 - 12 * q^87 - 24 * q^89 - 2 * q^91 - 8 * q^95 - 12 * q^97 - 8 * q^99

Introduction

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