LMFDB - Embedded newform 1568.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,4,Mod(1,1568)] 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("1568.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1568.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,6,0,0,0,38,0,44,0,-14,0,56,0,-96,0,-170,0,0,0,152,0, -158] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.5149948890$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54138$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.08276 q^{3} +9.08276 q^{5} -17.4966 q^{9} +34.1655 q^{11} -13.0828 q^{13} +28.0000 q^{15} +12.8276 q^{17} -139.745 q^{19} -21.3242 q^{23} -42.5034 q^{25} -137.172 q^{27} +142.814 q^{29} -192.152 q^{31} +105.324 q^{33} -115.834 q^{37} -40.3311 q^{39} -396.800 q^{41} +252.510 q^{43} -158.917 q^{45} -11.8208 q^{47} +39.5445 q^{51} -295.297 q^{53} +310.317 q^{55} -430.800 q^{57} -269.731 q^{59} +233.745 q^{61} -118.828 q^{65} +975.918 q^{67} -65.7374 q^{69} -870.317 q^{71} -391.600 q^{73} -131.028 q^{75} -897.573 q^{79} +49.5377 q^{81} +658.159 q^{83} +116.510 q^{85} +440.261 q^{87} -32.6210 q^{89} -592.358 q^{93} -1269.27 q^{95} -991.145 q^{97} -597.780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} + 38 q^{9} + 44 q^{11} - 14 q^{13} + 56 q^{15} - 96 q^{17} - 170 q^{19} + 152 q^{23} - 158 q^{25} - 396 q^{27} - 128 q^{29} - 68 q^{31} + 16 q^{33} - 256 q^{37} - 32 q^{39} - 88 q^{41}+ \cdots - 52 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 + 38 * q^9 + 44 * q^11 - 14 * q^13 + 56 * q^15 - 96 * q^17 - 170 * q^19 + 152 * q^23 - 158 * q^25 - 396 * q^27 - 128 * q^29 - 68 * q^31 + 16 * q^33 - 256 * q^37 - 32 * q^39 - 88 * q^41 + 724 * q^43 - 330 * q^45 + 244 * q^47 + 1028 * q^51 + 188 * q^53 + 280 * q^55 - 156 * q^57 - 138 * q^59 + 358 * q^61 - 116 * q^65 + 200 * q^67 - 1640 * q^69 - 1400 * q^71 + 628 * q^73 + 918 * q^75 + 200 * q^79 + 902 * q^81 - 618 * q^83 + 452 * q^85 + 2900 * q^87 + 908 * q^89 - 1720 * q^93 - 1176 * q^95 - 1520 * q^97 - 52 * 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
$3$	3.08276	0.593278	0.296639	−	0.954990i	$-0.404134\pi$
$3$	3.08276	0.593278	0.296639	+	0.954990i	$0.404134\pi$
$4$	0	0
$5$	9.08276	0.812387	0.406193	−	0.913787i	$-0.366856\pi$
$5$	9.08276	0.812387	0.406193	+	0.913787i	$0.366856\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−17.4966	−0.648021
$10$	0	0
$11$	34.1655	0.936481	0.468241	−	0.883601i	$-0.344888\pi$
$11$	34.1655	0.936481	0.468241	+	0.883601i	$0.344888\pi$
$12$	0	0
$13$	−13.0828	−0.279116	−0.139558	−	0.990214i	$-0.544568\pi$
$13$	−13.0828	−0.279116	−0.139558	+	0.990214i	$0.544568\pi$
$14$	0	0
$15$	28.0000	0.481971
$16$	0	0
$17$	12.8276	0.183009	0.0915046	−	0.995805i	$-0.470832\pi$
$17$	12.8276	0.183009	0.0915046	+	0.995805i	$0.470832\pi$
$18$	0	0
$19$	−139.745	−1.68735	−0.843676	−	0.536853i	$-0.819613\pi$
$19$	−139.745	−1.68735	−0.843676	+	0.536853i	$0.819613\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−21.3242	−0.193322	−0.0966609	−	0.995317i	$-0.530816\pi$
$23$	−21.3242	−0.193322	−0.0966609	+	0.995317i	$0.530816\pi$
$24$	0	0
$25$	−42.5034	−0.340027
$26$	0	0
$27$	−137.172	−0.977735
$28$	0	0
$29$	142.814	0.914479	0.457239	−	0.889344i	$-0.348838\pi$
$29$	142.814	0.914479	0.457239	+	0.889344i	$0.348838\pi$
$30$	0	0
$31$	−192.152	−1.11327	−0.556637	−	0.830756i	$-0.687909\pi$
$31$	−192.152	−1.11327	−0.556637	+	0.830756i	$0.687909\pi$
$32$	0	0
$33$	105.324	0.555594
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−115.834	−0.514678	−0.257339	−	0.966321i	$-0.582846\pi$
$37$	−115.834	−0.514678	−0.257339	+	0.966321i	$0.582846\pi$
$38$	0	0
$39$	−40.3311	−0.165593
$40$	0	0
$41$	−396.800	−1.51146	−0.755729	−	0.654884i	$-0.772717\pi$
$41$	−396.800	−1.51146	−0.755729	+	0.654884i	$0.772717\pi$
$42$	0	0
$43$	252.510	0.895522	0.447761	−	0.894153i	$-0.352222\pi$
$43$	252.510	0.895522	0.447761	+	0.894153i	$0.352222\pi$
$44$	0	0
$45$	−158.917	−0.526444
$46$	0	0
$47$	−11.8208	−0.0366859	−0.0183430	−	0.999832i	$-0.505839\pi$
$47$	−11.8208	−0.0366859	−0.0183430	+	0.999832i	$0.505839\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	39.5445	0.108575
$52$	0	0
$53$	−295.297	−0.765323	−0.382662	−	0.923889i	$-0.624992\pi$
$53$	−295.297	−0.765323	−0.382662	+	0.923889i	$0.624992\pi$
$54$	0	0
$55$	310.317	0.760785
$56$	0	0
$57$	−430.800	−1.00107
$58$	0	0
$59$	−269.731	−0.595187	−0.297593	−	0.954693i	$-0.596184\pi$
$59$	−269.731	−0.595187	−0.297593	+	0.954693i	$0.596184\pi$
$60$	0	0
$61$	233.745	0.490622	0.245311	−	0.969444i	$-0.421110\pi$
$61$	233.745	0.490622	0.245311	+	0.969444i	$0.421110\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−118.828	−0.226750
$66$	0	0
$67$	975.918	1.77951	0.889756	−	0.456436i	$-0.150874\pi$
$67$	975.918	1.77951	0.889756	+	0.456436i	$0.150874\pi$
$68$	0	0
$69$	−65.7374	−0.114694
$70$	0	0
$71$	−870.317	−1.45476	−0.727378	−	0.686237i	$-0.759261\pi$
$71$	−870.317	−1.45476	−0.727378	+	0.686237i	$0.759261\pi$
$72$	0	0
$73$	−391.600	−0.627854	−0.313927	−	0.949447i	$-0.601645\pi$
$73$	−391.600	−0.627854	−0.313927	+	0.949447i	$0.601645\pi$
$74$	0	0
$75$	−131.028	−0.201731
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−897.573	−1.27829	−0.639145	−	0.769087i	$-0.720711\pi$
$79$	−897.573	−1.27829	−0.639145	+	0.769087i	$0.720711\pi$
$80$	0	0
$81$	49.5377	0.0679529
$82$	0	0
$83$	658.159	0.870390	0.435195	−	0.900336i	$-0.356679\pi$
$83$	658.159	0.870390	0.435195	+	0.900336i	$0.356679\pi$
$84$	0	0
$85$	116.510	0.148674
$86$	0	0
$87$	440.261	0.542540
$88$	0	0
$89$	−32.6210	−0.0388519	−0.0194260	−	0.999811i	$-0.506184\pi$
$89$	−32.6210	−0.0388519	−0.0194260	+	0.999811i	$0.506184\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−592.358	−0.660481
$94$	0	0
$95$	−1269.27	−1.37078
$96$	0	0
$97$	−991.145	−1.03748	−0.518740	−	0.854932i	$-0.673599\pi$
$97$	−991.145	−1.03748	−0.518740	+	0.854932i	$0.673599\pi$
$98$	0	0
$99$	−597.780	−0.606860
$100$	0	0
$101$	−501.180	−0.493755	−0.246877	−	0.969047i	$-0.579405\pi$
$101$	−501.180	−0.493755	−0.246877	+	0.969047i	$0.579405\pi$
$102$	0	0
$103$	709.503	0.678733	0.339366	−	0.940654i	$-0.389787\pi$
$103$	709.503	0.678733	0.339366	+	0.940654i	$0.389787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−860.304	−0.777278	−0.388639	−	0.921390i	$-0.627055\pi$
$107$	−860.304	−0.777278	−0.388639	+	0.921390i	$0.627055\pi$
$108$	0	0
$109$	1415.72	1.24405	0.622027	−	0.782996i	$-0.286309\pi$
$109$	1415.72	1.24405	0.622027	+	0.782996i	$0.286309\pi$
$110$	0	0
$111$	−357.090	−0.305347
$112$	0	0
$113$	−1621.82	−1.35016	−0.675080	−	0.737745i	$-0.735891\pi$
$113$	−1621.82	−1.35016	−0.675080	+	0.737745i	$0.735891\pi$
$114$	0	0
$115$	−193.683	−0.157052
$116$	0	0
$117$	228.904	0.180873
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−163.717	−0.123003
$122$	0	0
$123$	−1223.24	−0.896715
$124$	0	0
$125$	−1521.39	−1.08862
$126$	0	0
$127$	1964.91	1.37289	0.686447	−	0.727179i	$-0.259169\pi$
$127$	1964.91	1.37289	0.686447	+	0.727179i	$0.259169\pi$
$128$	0	0
$129$	778.429	0.531294
$130$	0	0
$131$	−1023.97	−0.682934	−0.341467	−	0.939894i	$-0.610924\pi$
$131$	−1023.97	−0.682934	−0.341467	+	0.939894i	$0.610924\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1245.90	−0.794299
$136$	0	0
$137$	2534.94	1.58084	0.790418	−	0.612568i	$-0.209864\pi$
$137$	2534.94	1.58084	0.790418	+	0.612568i	$0.209864\pi$
$138$	0	0
$139$	509.676	0.311009	0.155504	−	0.987835i	$-0.450300\pi$
$139$	509.676	0.311009	0.155504	+	0.987835i	$0.450300\pi$
$140$	0	0
$141$	−36.4406	−0.0217649
$142$	0	0
$143$	−446.979	−0.261387
$144$	0	0
$145$	1297.14	0.742911
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2263.55	−1.24454	−0.622271	−	0.782802i	$-0.713790\pi$
$149$	−2263.55	−1.24454	−0.622271	+	0.782802i	$0.713790\pi$
$150$	0	0
$151$	1184.44	0.638334	0.319167	−	0.947699i	$-0.396597\pi$
$151$	1184.44	0.638334	0.319167	+	0.947699i	$0.396597\pi$
$152$	0	0
$153$	−224.440	−0.118594
$154$	0	0
$155$	−1745.27	−0.904409
$156$	0	0
$157$	1426.70	0.725241	0.362621	−	0.931937i	$-0.381882\pi$
$157$	1426.70	0.725241	0.362621	+	0.931937i	$0.381882\pi$
$158$	0	0
$159$	−910.330	−0.454049
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2473.04	1.18836	0.594181	−	0.804331i	$-0.297476\pi$
$163$	2473.04	1.18836	0.594181	+	0.804331i	$0.297476\pi$
$164$	0	0
$165$	956.635	0.451357
$166$	0	0
$167$	−338.083	−0.156657	−0.0783284	−	0.996928i	$-0.524958\pi$
$167$	−338.083	−0.156657	−0.0783284	+	0.996928i	$0.524958\pi$
$168$	0	0
$169$	−2025.84	−0.922094
$170$	0	0
$171$	2445.06	1.09344
$172$	0	0
$173$	−967.482	−0.425181	−0.212591	−	0.977141i	$-0.568190\pi$
$173$	−967.482	−0.425181	−0.212591	+	0.977141i	$0.568190\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−831.517	−0.353111
$178$	0	0
$179$	2438.13	1.01807	0.509034	−	0.860747i	$-0.330003\pi$
$179$	2438.13	1.01807	0.509034	+	0.860747i	$0.330003\pi$
$180$	0	0
$181$	−4711.93	−1.93500	−0.967499	−	0.252875i	$-0.918624\pi$
$181$	−4711.93	−1.93500	−0.967499	+	0.252875i	$0.918624\pi$
$182$	0	0
$183$	720.580	0.291075
$184$	0	0
$185$	−1052.10	−0.418117
$186$	0	0
$187$	438.263	0.171385
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5172.66	1.95959	0.979793	−	0.200015i	$-0.0640991\pi$
$191$	5172.66	1.95959	0.979793	+	0.200015i	$0.0640991\pi$
$192$	0	0
$193$	2199.06	0.820166	0.410083	−	0.912048i	$-0.365500\pi$
$193$	2199.06	0.820166	0.410083	+	0.912048i	$0.365500\pi$
$194$	0	0
$195$	−366.317	−0.134526
$196$	0	0
$197$	−3572.73	−1.29212	−0.646058	−	0.763289i	$-0.723583\pi$
$197$	−3572.73	−1.29212	−0.646058	+	0.763289i	$0.723583\pi$
$198$	0	0
$199$	−3198.17	−1.13926	−0.569628	−	0.821903i	$-0.692913\pi$
$199$	−3198.17	−1.13926	−0.569628	+	0.821903i	$0.692913\pi$
$200$	0	0
$201$	3008.52	1.05575
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3604.04	−1.22789
$206$	0	0
$207$	373.100	0.125277
$208$	0	0
$209$	−4774.46	−1.58017
$210$	0	0
$211$	2686.70	0.876589	0.438294	−	0.898831i	$-0.355583\pi$
$211$	2686.70	0.876589	0.438294	+	0.898831i	$0.355583\pi$
$212$	0	0
$213$	−2682.98	−0.863075
$214$	0	0
$215$	2293.49	0.727511
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1207.21	−0.372492
$220$	0	0
$221$	−167.821	−0.0510808
$222$	0	0
$223$	−6352.75	−1.90767	−0.953837	−	0.300324i	$-0.902905\pi$
$223$	−6352.75	−1.90767	−0.953837	+	0.300324i	$0.902905\pi$
$224$	0	0
$225$	743.664	0.220345
$226$	0	0
$227$	−1205.23	−0.352397	−0.176199	−	0.984355i	$-0.556380\pi$
$227$	−1205.23	−0.352397	−0.176199	+	0.984355i	$0.556380\pi$
$228$	0	0
$229$	−867.345	−0.250287	−0.125144	−	0.992139i	$-0.539939\pi$
$229$	−867.345	−0.250287	−0.125144	+	0.992139i	$0.539939\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−232.151	−0.0652734	−0.0326367	−	0.999467i	$-0.510390\pi$
$233$	−232.151	−0.0652734	−0.0326367	+	0.999467i	$0.510390\pi$
$234$	0	0
$235$	−107.365	−0.0298031
$236$	0	0
$237$	−2767.00	−0.758381
$238$	0	0
$239$	131.283	0.0355314	0.0177657	−	0.999842i	$-0.494345\pi$
$239$	131.283	0.0355314	0.0177657	+	0.999842i	$0.494345\pi$
$240$	0	0
$241$	−5858.79	−1.56597	−0.782983	−	0.622043i	$-0.786303\pi$
$241$	−5858.79	−1.56597	−0.782983	+	0.622043i	$0.786303\pi$
$242$	0	0
$243$	3856.37	1.01805
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1828.25	0.470966
$248$	0	0
$249$	2028.95	0.516383
$250$	0	0
$251$	1795.55	0.451531	0.225765	−	0.974182i	$-0.427512\pi$
$251$	1795.55	0.451531	0.225765	+	0.974182i	$0.427512\pi$
$252$	0	0
$253$	−728.553	−0.181042
$254$	0	0
$255$	359.174	0.0882052
$256$	0	0
$257$	7753.70	1.88195	0.940977	−	0.338469i	$-0.109909\pi$
$257$	7753.70	1.88195	0.940977	+	0.338469i	$0.109909\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2498.75	−0.592602
$262$	0	0
$263$	−5937.02	−1.39199	−0.695993	−	0.718048i	$-0.745036\pi$
$263$	−5937.02	−1.39199	−0.695993	+	0.718048i	$0.745036\pi$
$264$	0	0
$265$	−2682.11	−0.621739
$266$	0	0
$267$	−100.563	−0.0230500
$268$	0	0
$269$	2313.02	0.524264	0.262132	−	0.965032i	$-0.415574\pi$
$269$	2313.02	0.524264	0.262132	+	0.965032i	$0.415574\pi$
$270$	0	0
$271$	−5212.08	−1.16831	−0.584154	−	0.811643i	$-0.698574\pi$
$271$	−5212.08	−1.16831	−0.584154	+	0.811643i	$0.698574\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1452.15	−0.318429
$276$	0	0
$277$	−6984.74	−1.51506	−0.757531	−	0.652799i	$-0.773594\pi$
$277$	−6984.74	−1.51506	−0.757531	+	0.652799i	$0.773594\pi$
$278$	0	0
$279$	3362.00	0.721425
$280$	0	0
$281$	−6845.51	−1.45327	−0.726635	−	0.687024i	$-0.758917\pi$
$281$	−6845.51	−1.45327	−0.726635	+	0.687024i	$0.758917\pi$
$282$	0	0
$283$	7726.11	1.62286	0.811430	−	0.584450i	$-0.198690\pi$
$283$	7726.11	1.62286	0.811430	+	0.584450i	$0.198690\pi$
$284$	0	0
$285$	−3912.86	−0.813255
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4748.45	−0.966508
$290$	0	0
$291$	−3055.46	−0.615514
$292$	0	0
$293$	−9674.17	−1.92891	−0.964456	−	0.264243i	$-0.914878\pi$
$293$	−9674.17	−1.92891	−0.964456	+	0.264243i	$0.914878\pi$
$294$	0	0
$295$	−2449.90	−0.483522
$296$	0	0
$297$	−4686.57	−0.915630
$298$	0	0
$299$	278.979	0.0539592
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1545.02	−0.292934
$304$	0	0
$305$	2123.05	0.398575
$306$	0	0
$307$	−2131.55	−0.396267	−0.198134	−	0.980175i	$-0.563488\pi$
$307$	−2131.55	−0.396267	−0.198134	+	0.980175i	$0.563488\pi$
$308$	0	0
$309$	2187.23	0.402677
$310$	0	0
$311$	6502.68	1.18564	0.592818	−	0.805336i	$-0.298015\pi$
$311$	6502.68	1.18564	0.592818	+	0.805336i	$0.298015\pi$
$312$	0	0
$313$	8879.89	1.60358	0.801791	−	0.597604i	$-0.203881\pi$
$313$	8879.89	1.60358	0.801791	+	0.597604i	$0.203881\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2818.86	−0.499441	−0.249720	−	0.968318i	$-0.580339\pi$
$317$	−2818.86	−0.499441	−0.249720	+	0.968318i	$0.580339\pi$
$318$	0	0
$319$	4879.31	0.856392
$320$	0	0
$321$	−2652.11	−0.461142
$322$	0	0
$323$	−1792.59	−0.308801
$324$	0	0
$325$	556.062	0.0949070
$326$	0	0
$327$	4364.34	0.738070
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5572.13	0.925292	0.462646	−	0.886543i	$-0.346900\pi$
$331$	5572.13	0.925292	0.462646	+	0.886543i	$0.346900\pi$
$332$	0	0
$333$	2026.71	0.333522
$334$	0	0
$335$	8864.03	1.44565
$336$	0	0
$337$	1014.24	0.163944	0.0819718	−	0.996635i	$-0.473878\pi$
$337$	1014.24	0.163944	0.0819718	+	0.996635i	$0.473878\pi$
$338$	0	0
$339$	−4999.69	−0.801020
$340$	0	0
$341$	−6564.97	−1.04256
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−597.078	−0.0931756
$346$	0	0
$347$	−863.408	−0.133574	−0.0667870	−	0.997767i	$-0.521275\pi$
$347$	−863.408	−0.133574	−0.0667870	+	0.997767i	$0.521275\pi$
$348$	0	0
$349$	−2772.92	−0.425303	−0.212652	−	0.977128i	$-0.568210\pi$
$349$	−2772.92	−0.425303	−0.212652	+	0.977128i	$0.568210\pi$
$350$	0	0
$351$	1794.59	0.272901
$352$	0	0
$353$	350.333	0.0528226	0.0264113	−	0.999651i	$-0.491592\pi$
$353$	350.333	0.0528226	0.0264113	+	0.999651i	$0.491592\pi$
$354$	0	0
$355$	−7904.89	−1.18182
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12932.4	−1.90124	−0.950618	−	0.310362i	$-0.899550\pi$
$359$	−12932.4	−1.90124	−0.950618	+	0.310362i	$0.899550\pi$
$360$	0	0
$361$	12669.6	1.84715
$362$	0	0
$363$	−504.700	−0.0729749
$364$	0	0
$365$	−3556.81	−0.510061
$366$	0	0
$367$	−969.349	−0.137874	−0.0689368	−	0.997621i	$-0.521961\pi$
$367$	−969.349	−0.137874	−0.0689368	+	0.997621i	$0.521961\pi$
$368$	0	0
$369$	6942.64	0.979457
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5399.55	−0.749539	−0.374769	−	0.927118i	$-0.622278\pi$
$373$	−5399.55	−0.749539	−0.374769	+	0.927118i	$0.622278\pi$
$374$	0	0
$375$	−4690.10	−0.645855
$376$	0	0
$377$	−1868.40	−0.255245
$378$	0	0
$379$	−2296.87	−0.311299	−0.155650	−	0.987812i	$-0.549747\pi$
$379$	−2296.87	−0.311299	−0.155650	+	0.987812i	$0.549747\pi$
$380$	0	0
$381$	6057.35	0.814508
$382$	0	0
$383$	9908.90	1.32199	0.660993	−	0.750392i	$-0.270135\pi$
$383$	9908.90	1.32199	0.660993	+	0.750392i	$0.270135\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4418.07	−0.580317
$388$	0	0
$389$	−3858.40	−0.502901	−0.251451	−	0.967870i	$-0.580908\pi$
$389$	−3858.40	−0.502901	−0.251451	+	0.967870i	$0.580908\pi$
$390$	0	0
$391$	−273.539	−0.0353797
$392$	0	0
$393$	−3156.64	−0.405170
$394$	0	0
$395$	−8152.44	−1.03847
$396$	0	0
$397$	−8571.26	−1.08357	−0.541787	−	0.840516i	$-0.682252\pi$
$397$	−8571.26	−1.08357	−0.541787	+	0.840516i	$0.682252\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8756.14	1.09043	0.545213	−	0.838298i	$-0.316449\pi$
$401$	8756.14	1.09043	0.545213	+	0.838298i	$0.316449\pi$
$402$	0	0
$403$	2513.88	0.310732
$404$	0	0
$405$	449.939	0.0552041
$406$	0	0
$407$	−3957.55	−0.481986
$408$	0	0
$409$	1739.67	0.210321	0.105160	−	0.994455i	$-0.466464\pi$
$409$	1739.67	0.210321	0.105160	+	0.994455i	$0.466464\pi$
$410$	0	0
$411$	7814.61	0.937875
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5977.90	0.707093
$416$	0	0
$417$	1571.21	0.184514
$418$	0	0
$419$	−7998.16	−0.932543	−0.466272	−	0.884642i	$-0.654403\pi$
$419$	−7998.16	−0.932543	−0.466272	+	0.884642i	$0.654403\pi$
$420$	0	0
$421$	8002.03	0.926354	0.463177	−	0.886266i	$-0.346709\pi$
$421$	8002.03	0.926354	0.463177	+	0.886266i	$0.346709\pi$
$422$	0	0
$423$	206.823	0.0237732
$424$	0	0
$425$	−545.218	−0.0622281
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1377.93	−0.155075
$430$	0	0
$431$	8374.93	0.935977	0.467989	−	0.883734i	$-0.344979\pi$
$431$	8374.93	0.935977	0.467989	+	0.883734i	$0.344979\pi$
$432$	0	0
$433$	10136.5	1.12501	0.562505	−	0.826794i	$-0.309838\pi$
$433$	10136.5	1.12501	0.562505	+	0.826794i	$0.309838\pi$
$434$	0	0
$435$	3998.79	0.440752
$436$	0	0
$437$	2979.95	0.326202
$438$	0	0
$439$	−11505.6	−1.25087	−0.625433	−	0.780278i	$-0.715078\pi$
$439$	−11505.6	−1.25087	−0.625433	+	0.780278i	$0.715078\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3045.88	0.326668	0.163334	−	0.986571i	$-0.447775\pi$
$443$	3045.88	0.326668	0.163334	+	0.986571i	$0.447775\pi$
$444$	0	0
$445$	−296.289	−0.0315628
$446$	0	0
$447$	−6977.97	−0.738360
$448$	0	0
$449$	−8831.38	−0.928238	−0.464119	−	0.885773i	$-0.653629\pi$
$449$	−8831.38	−0.928238	−0.464119	+	0.885773i	$0.653629\pi$
$450$	0	0
$451$	−13556.9	−1.41545
$452$	0	0
$453$	3651.35	0.378709
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4058.54	0.415428	0.207714	−	0.978190i	$-0.433398\pi$
$457$	4058.54	0.415428	0.207714	+	0.978190i	$0.433398\pi$
$458$	0	0
$459$	−1759.60	−0.178934
$460$	0	0
$461$	6858.56	0.692917	0.346459	−	0.938065i	$-0.387384\pi$
$461$	6858.56	0.692917	0.346459	+	0.938065i	$0.387384\pi$
$462$	0	0
$463$	−3587.32	−0.360079	−0.180040	−	0.983659i	$-0.557623\pi$
$463$	−3587.32	−0.360079	−0.180040	+	0.983659i	$0.557623\pi$
$464$	0	0
$465$	−5380.25	−0.536566
$466$	0	0
$467$	−12553.5	−1.24391	−0.621954	−	0.783054i	$-0.713661\pi$
$467$	−12553.5	−1.24391	−0.621954	+	0.783054i	$0.713661\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4398.17	0.430270
$472$	0	0
$473$	8627.15	0.838640
$474$	0	0
$475$	5939.64	0.573746
$476$	0	0
$477$	5166.68	0.495946
$478$	0	0
$479$	7622.42	0.727092	0.363546	−	0.931576i	$-0.381566\pi$
$479$	7622.42	0.727092	0.363546	+	0.931576i	$0.381566\pi$
$480$	0	0
$481$	1515.43	0.143655
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9002.33	−0.842835
$486$	0	0
$487$	19040.9	1.77171	0.885856	−	0.463960i	$-0.153572\pi$
$487$	19040.9	1.77171	0.885856	+	0.463960i	$0.153572\pi$
$488$	0	0
$489$	7623.78	0.705029
$490$	0	0
$491$	14451.3	1.32827	0.664134	−	0.747613i	$-0.268800\pi$
$491$	14451.3	1.32827	0.664134	+	0.747613i	$0.268800\pi$
$492$	0	0
$493$	1831.96	0.167358
$494$	0	0
$495$	−5429.49	−0.493005
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8904.70	−0.798856	−0.399428	−	0.916765i	$-0.630791\pi$
$499$	−8904.70	−0.798856	−0.399428	+	0.916765i	$0.630791\pi$
$500$	0	0
$501$	−1042.23	−0.0929410
$502$	0	0
$503$	14787.6	1.31083	0.655413	−	0.755271i	$-0.272495\pi$
$503$	14787.6	1.31083	0.655413	+	0.755271i	$0.272495\pi$
$504$	0	0
$505$	−4552.10	−0.401120
$506$	0	0
$507$	−6245.19	−0.547058
$508$	0	0
$509$	−20698.0	−1.80240	−0.901201	−	0.433401i	$-0.857313\pi$
$509$	−20698.0	−1.80240	−0.901201	+	0.433401i	$0.857313\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	19169.1	1.64978
$514$	0	0
$515$	6444.25	0.551393
$516$	0	0
$517$	−403.863	−0.0343557
$518$	0	0
$519$	−2982.52	−0.252251
$520$	0	0
$521$	5283.86	0.444319	0.222159	−	0.975010i	$-0.428689\pi$
$521$	5283.86	0.444319	0.222159	+	0.975010i	$0.428689\pi$
$522$	0	0
$523$	1489.36	0.124522	0.0622611	−	0.998060i	$-0.480169\pi$
$523$	1489.36	0.124522	0.0622611	+	0.998060i	$0.480169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2464.85	−0.203739
$528$	0	0
$529$	−11712.3	−0.962627
$530$	0	0
$531$	4719.37	0.385694
$532$	0	0
$533$	5191.24	0.421872
$534$	0	0
$535$	−7813.93	−0.631450
$536$	0	0
$537$	7516.16	0.603997
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5574.20	0.442982	0.221491	−	0.975162i	$-0.428908\pi$
$541$	5574.20	0.442982	0.221491	+	0.975162i	$0.428908\pi$
$542$	0	0
$543$	−14525.7	−1.14799
$544$	0	0
$545$	12858.7	1.01065
$546$	0	0
$547$	809.449	0.0632715	0.0316358	−	0.999499i	$-0.489928\pi$
$547$	809.449	0.0632715	0.0316358	+	0.999499i	$0.489928\pi$
$548$	0	0
$549$	−4089.73	−0.317934
$550$	0	0
$551$	−19957.5	−1.54305
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3243.37	−0.248060
$556$	0	0
$557$	−12199.2	−0.927998	−0.463999	−	0.885836i	$-0.653586\pi$
$557$	−12199.2	−0.927998	−0.463999	+	0.885836i	$0.653586\pi$
$558$	0	0
$559$	−3303.53	−0.249954
$560$	0	0
$561$	1351.06	0.101679
$562$	0	0
$563$	22708.9	1.69994	0.849971	−	0.526829i	$-0.176619\pi$
$563$	22708.9	1.69994	0.849971	+	0.526829i	$0.176619\pi$
$564$	0	0
$565$	−14730.6	−1.09685
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19186.3	1.41359	0.706793	−	0.707421i	$-0.250141\pi$
$569$	19186.3	1.41359	0.706793	+	0.707421i	$0.250141\pi$
$570$	0	0
$571$	−15757.5	−1.15487	−0.577433	−	0.816438i	$-0.695946\pi$
$571$	−15757.5	−1.15487	−0.577433	+	0.816438i	$0.695946\pi$
$572$	0	0
$573$	15946.1	1.16258
$574$	0	0
$575$	906.352	0.0657347
$576$	0	0
$577$	−4445.32	−0.320729	−0.160365	−	0.987058i	$-0.551267\pi$
$577$	−4445.32	−0.320729	−0.160365	+	0.987058i	$0.551267\pi$
$578$	0	0
$579$	6779.19	0.486586
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10089.0	−0.716711
$584$	0	0
$585$	2079.08	0.146939
$586$	0	0
$587$	−23183.2	−1.63011	−0.815054	−	0.579385i	$-0.803293\pi$
$587$	−23183.2	−1.63011	−0.815054	+	0.579385i	$0.803293\pi$
$588$	0	0
$589$	26852.2	1.87848
$590$	0	0
$591$	−11013.9	−0.766583
$592$	0	0
$593$	9639.30	0.667519	0.333759	−	0.942658i	$-0.391683\pi$
$593$	9639.30	0.667519	0.333759	+	0.942658i	$0.391683\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9859.18	−0.675895
$598$	0	0
$599$	1294.46	0.0882976	0.0441488	−	0.999025i	$-0.485942\pi$
$599$	1294.46	0.0882976	0.0441488	+	0.999025i	$0.485942\pi$
$600$	0	0
$601$	3912.65	0.265558	0.132779	−	0.991146i	$-0.457610\pi$
$601$	3912.65	0.265558	0.132779	+	0.991146i	$0.457610\pi$
$602$	0	0
$603$	−17075.2	−1.15316
$604$	0	0
$605$	−1487.00	−0.0999260
$606$	0	0
$607$	20986.8	1.40334	0.701670	−	0.712503i	$-0.252438\pi$
$607$	20986.8	1.40334	0.701670	+	0.712503i	$0.252438\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	154.648	0.0102396
$612$	0	0
$613$	−14701.1	−0.968633	−0.484316	−	0.874893i	$-0.660932\pi$
$613$	−14701.1	−0.968633	−0.484316	+	0.874893i	$0.660932\pi$
$614$	0	0
$615$	−11110.4	−0.728479
$616$	0	0
$617$	−3795.09	−0.247625	−0.123812	−	0.992306i	$-0.539512\pi$
$617$	−3795.09	−0.247625	−0.123812	+	0.992306i	$0.539512\pi$
$618$	0	0
$619$	11335.1	0.736020	0.368010	−	0.929822i	$-0.380039\pi$
$619$	11335.1	0.736020	0.368010	+	0.929822i	$0.380039\pi$
$620$	0	0
$621$	2925.09	0.189017
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8505.53	−0.544354
$626$	0	0
$627$	−14718.5	−0.937482
$628$	0	0
$629$	−1485.88	−0.0941907
$630$	0	0
$631$	17565.2	1.10817	0.554087	−	0.832459i	$-0.313067\pi$
$631$	17565.2	1.10817	0.554087	+	0.832459i	$0.313067\pi$
$632$	0	0
$633$	8282.47	0.520061
$634$	0	0
$635$	17846.8	1.11532
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15227.6	0.942713
$640$	0	0
$641$	−10053.0	−0.619451	−0.309726	−	0.950826i	$-0.600237\pi$
$641$	−10053.0	−0.619451	−0.309726	+	0.950826i	$0.600237\pi$
$642$	0	0
$643$	26476.7	1.62386	0.811928	−	0.583757i	$-0.198418\pi$
$643$	26476.7	1.62386	0.811928	+	0.583757i	$0.198418\pi$
$644$	0	0
$645$	7070.29	0.431616
$646$	0	0
$647$	−4372.68	−0.265700	−0.132850	−	0.991136i	$-0.542413\pi$
$647$	−4372.68	−0.265700	−0.132850	+	0.991136i	$0.542413\pi$
$648$	0	0
$649$	−9215.51	−0.557381
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18328.4	1.09839	0.549193	−	0.835696i	$-0.314935\pi$
$653$	18328.4	1.09839	0.549193	+	0.835696i	$0.314935\pi$
$654$	0	0
$655$	−9300.44	−0.554807
$656$	0	0
$657$	6851.67	0.406863
$658$	0	0
$659$	−19536.1	−1.15481	−0.577404	−	0.816458i	$-0.695934\pi$
$659$	−19536.1	−1.15481	−0.577404	+	0.816458i	$0.695934\pi$
$660$	0	0
$661$	23169.3	1.36336	0.681679	−	0.731651i	$-0.261250\pi$
$661$	23169.3	1.36336	0.681679	+	0.731651i	$0.261250\pi$
$662$	0	0
$663$	−517.352	−0.0303051
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3045.39	−0.176789
$668$	0	0
$669$	−19584.0	−1.13178
$670$	0	0
$671$	7986.02	0.459459
$672$	0	0
$673$	11370.9	0.651287	0.325644	−	0.945493i	$-0.394419\pi$
$673$	11370.9	0.651287	0.325644	+	0.945493i	$0.394419\pi$
$674$	0	0
$675$	5830.30	0.332457
$676$	0	0
$677$	3556.00	0.201873	0.100937	−	0.994893i	$-0.467816\pi$
$677$	3556.00	0.201873	0.100937	+	0.994893i	$0.467816\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3715.45	−0.209069
$682$	0	0
$683$	16124.8	0.903367	0.451683	−	0.892178i	$-0.350824\pi$
$683$	16124.8	0.903367	0.451683	+	0.892178i	$0.350824\pi$
$684$	0	0
$685$	23024.2	1.28425
$686$	0	0
$687$	−2673.82	−0.148490
$688$	0	0
$689$	3863.30	0.213614
$690$	0	0
$691$	14358.8	0.790499	0.395250	−	0.918574i	$-0.370658\pi$
$691$	14358.8	0.790499	0.395250	+	0.918574i	$0.370658\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4629.27	0.252659
$696$	0	0
$697$	−5090.00	−0.276611
$698$	0	0
$699$	−715.665	−0.0387253
$700$	0	0
$701$	16098.4	0.867374	0.433687	−	0.901064i	$-0.357212\pi$
$701$	16098.4	0.867374	0.433687	+	0.901064i	$0.357212\pi$
$702$	0	0
$703$	16187.3	0.868442
$704$	0	0
$705$	−330.982	−0.0176815
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6849.63	0.362826	0.181413	−	0.983407i	$-0.441933\pi$
$709$	6849.63	0.362826	0.181413	+	0.983407i	$0.441933\pi$
$710$	0	0
$711$	15704.5	0.828359
$712$	0	0
$713$	4097.48	0.215220
$714$	0	0
$715$	−4059.81	−0.212347
$716$	0	0
$717$	404.715	0.0210800
$718$	0	0
$719$	9257.56	0.480179	0.240090	−	0.970751i	$-0.422823\pi$
$719$	9257.56	0.480179	0.240090	+	0.970751i	$0.422823\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18061.3	−0.929053
$724$	0	0
$725$	−6070.08	−0.310948
$726$	0	0
$727$	7188.96	0.366745	0.183373	−	0.983043i	$-0.441299\pi$
$727$	7188.96	0.366745	0.183373	+	0.983043i	$0.441299\pi$
$728$	0	0
$729$	10550.7	0.536033
$730$	0	0
$731$	3239.11	0.163889
$732$	0	0
$733$	−19359.3	−0.975515	−0.487757	−	0.872979i	$-0.662185\pi$
$733$	−19359.3	−0.975515	−0.487757	+	0.872979i	$0.662185\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33342.7	1.66648
$738$	0	0
$739$	−1159.95	−0.0577395	−0.0288697	−	0.999583i	$-0.509191\pi$
$739$	−1159.95	−0.0577395	−0.0288697	+	0.999583i	$0.509191\pi$
$740$	0	0
$741$	5636.06	0.279414
$742$	0	0
$743$	−12095.9	−0.597248	−0.298624	−	0.954371i	$-0.596528\pi$
$743$	−12095.9	−0.597248	−0.298624	+	0.954371i	$0.596528\pi$
$744$	0	0
$745$	−20559.2	−1.01105
$746$	0	0
$747$	−11515.5	−0.564031
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22296.3	1.08336	0.541679	−	0.840585i	$-0.317789\pi$
$751$	22296.3	1.08336	0.541679	+	0.840585i	$0.317789\pi$
$752$	0	0
$753$	5535.26	0.267883
$754$	0	0
$755$	10758.0	0.518574
$756$	0	0
$757$	17611.2	0.845563	0.422782	−	0.906232i	$-0.361054\pi$
$757$	17611.2	0.845563	0.422782	+	0.906232i	$0.361054\pi$
$758$	0	0
$759$	−2245.95	−0.107408
$760$	0	0
$761$	28792.7	1.37153	0.685765	−	0.727823i	$-0.259468\pi$
$761$	28792.7	1.37153	0.685765	+	0.727823i	$0.259468\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2038.53	−0.0963441
$766$	0	0
$767$	3528.83	0.166126
$768$	0	0
$769$	34104.7	1.59928	0.799640	−	0.600480i	$-0.205024\pi$
$769$	34104.7	1.59928	0.799640	+	0.600480i	$0.205024\pi$
$770$	0	0
$771$	23902.8	1.11652
$772$	0	0
$773$	−3821.35	−0.177807	−0.0889033	−	0.996040i	$-0.528336\pi$
$773$	−3821.35	−0.177807	−0.0889033	+	0.996040i	$0.528336\pi$
$774$	0	0
$775$	8167.11	0.378544
$776$	0	0
$777$	0	0
$778$	0	0
$779$	55450.8	2.55036
$780$	0	0
$781$	−29734.8	−1.36235
$782$	0	0
$783$	−19590.1	−0.894117
$784$	0	0
$785$	12958.4	0.589177
$786$	0	0
$787$	−3580.88	−0.162191	−0.0810956	−	0.996706i	$-0.525842\pi$
$787$	−3580.88	−0.162191	−0.0810956	+	0.996706i	$0.525842\pi$
$788$	0	0
$789$	−18302.4	−0.825835
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3058.03	−0.136940
$794$	0	0
$795$	−8268.31	−0.368864
$796$	0	0
$797$	31970.7	1.42091	0.710453	−	0.703745i	$-0.248490\pi$
$797$	31970.7	1.42091	0.710453	+	0.703745i	$0.248490\pi$
$798$	0	0
$799$	−151.632	−0.00671386
$800$	0	0
$801$	570.756	0.0251769
$802$	0	0
$803$	−13379.2	−0.587974
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7130.48	0.311034
$808$	0	0
$809$	5730.20	0.249027	0.124514	−	0.992218i	$-0.460263\pi$
$809$	5730.20	0.249027	0.124514	+	0.992218i	$0.460263\pi$
$810$	0	0
$811$	−16525.5	−0.715523	−0.357761	−	0.933813i	$-0.616460\pi$
$811$	−16525.5	−0.715523	−0.357761	+	0.933813i	$0.616460\pi$
$812$	0	0
$813$	−16067.6	−0.693131
$814$	0	0
$815$	22462.0	0.965410
$816$	0	0
$817$	−35287.0	−1.51106
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9304.93	0.395547	0.197774	−	0.980248i	$-0.436629\pi$
$821$	9304.93	0.395547	0.197774	+	0.980248i	$0.436629\pi$
$822$	0	0
$823$	−20220.8	−0.856443	−0.428222	−	0.903674i	$-0.640860\pi$
$823$	−20220.8	−0.856443	−0.428222	+	0.903674i	$0.640860\pi$
$824$	0	0
$825$	−4476.64	−0.188917
$826$	0	0
$827$	37983.5	1.59712	0.798558	−	0.601919i	$-0.205597\pi$
$827$	37983.5	1.59712	0.798558	+	0.601919i	$0.205597\pi$
$828$	0	0
$829$	24674.8	1.03376	0.516882	−	0.856056i	$-0.327092\pi$
$829$	24674.8	1.03376	0.516882	+	0.856056i	$0.327092\pi$
$830$	0	0
$831$	−21532.3	−0.898853
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3070.73	−0.127266
$836$	0	0
$837$	26357.9	1.08849
$838$	0	0
$839$	−41574.9	−1.71076	−0.855378	−	0.518005i	$-0.826675\pi$
$839$	−41574.9	−1.71076	−0.855378	+	0.518005i	$0.826675\pi$
$840$	0	0
$841$	−3993.18	−0.163729
$842$	0	0
$843$	−21103.1	−0.862193
$844$	0	0
$845$	−18400.2	−0.749097
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23817.8	0.962807
$850$	0	0
$851$	2470.08	0.0994984
$852$	0	0
$853$	−13103.0	−0.525953	−0.262976	−	0.964802i	$-0.584704\pi$
$853$	−13103.0	−0.525953	−0.262976	+	0.964802i	$0.584704\pi$
$854$	0	0
$855$	22207.9	0.888296
$856$	0	0
$857$	4100.11	0.163427	0.0817135	−	0.996656i	$-0.473961\pi$
$857$	4100.11	0.163427	0.0817135	+	0.996656i	$0.473961\pi$
$858$	0	0
$859$	−17741.5	−0.704695	−0.352347	−	0.935869i	$-0.614616\pi$
$859$	−17741.5	−0.704695	−0.352347	+	0.935869i	$0.614616\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41942.8	−1.65440	−0.827201	−	0.561907i	$-0.810068\pi$
$863$	−41942.8	−1.65440	−0.827201	+	0.561907i	$0.810068\pi$
$864$	0	0
$865$	−8787.41	−0.345412
$866$	0	0
$867$	−14638.3	−0.573408
$868$	0	0
$869$	−30666.1	−1.19709
$870$	0	0
$871$	−12767.7	−0.496690
$872$	0	0
$873$	17341.6	0.672309
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27562.4	1.06125	0.530625	−	0.847607i	$-0.321957\pi$
$877$	27562.4	1.06125	0.530625	+	0.847607i	$0.321957\pi$
$878$	0	0
$879$	−29823.2	−1.14438
$880$	0	0
$881$	39863.8	1.52446	0.762228	−	0.647308i	$-0.224105\pi$
$881$	39863.8	1.52446	0.762228	+	0.647308i	$0.224105\pi$
$882$	0	0
$883$	−38030.3	−1.44940	−0.724701	−	0.689064i	$-0.758022\pi$
$883$	−38030.3	−1.44940	−0.724701	+	0.689064i	$0.758022\pi$
$884$	0	0
$885$	−7552.47	−0.286863
$886$	0	0
$887$	12130.5	0.459190	0.229595	−	0.973286i	$-0.426260\pi$
$887$	12130.5	0.459190	0.229595	+	0.973286i	$0.426260\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1692.48	0.0636366
$892$	0	0
$893$	1651.89	0.0619020
$894$	0	0
$895$	22144.9	0.827065
$896$	0	0
$897$	860.027	0.0320128
$898$	0	0
$899$	−27442.0	−1.01807
$900$	0	0
$901$	−3787.96	−0.140061
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−42797.3	−1.57197
$906$	0	0
$907$	42557.0	1.55797	0.778987	−	0.627039i	$-0.215734\pi$
$907$	42557.0	1.55797	0.778987	+	0.627039i	$0.215734\pi$
$908$	0	0
$909$	8768.93	0.319964
$910$	0	0
$911$	9996.30	0.363548	0.181774	−	0.983340i	$-0.441816\pi$
$911$	9996.30	0.363548	0.181774	+	0.983340i	$0.441816\pi$
$912$	0	0
$913$	22486.4	0.815104
$914$	0	0
$915$	6544.86	0.236466
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8273.88	−0.296986	−0.148493	−	0.988913i	$-0.547442\pi$
$919$	−8273.88	−0.296986	−0.148493	+	0.988913i	$0.547442\pi$
$920$	0	0
$921$	−6571.06	−0.235097
$922$	0	0
$923$	11386.2	0.406045
$924$	0	0
$925$	4923.36	0.175004
$926$	0	0
$927$	−12413.9	−0.439833
$928$	0	0
$929$	−8388.40	−0.296248	−0.148124	−	0.988969i	$-0.547323\pi$
$929$	−8388.40	−0.296248	−0.148124	+	0.988969i	$0.547323\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20046.2	0.703412
$934$	0	0
$935$	3980.63	0.139231
$936$	0	0
$937$	3757.33	0.130999	0.0654997	−	0.997853i	$-0.479136\pi$
$937$	3757.33	0.130999	0.0654997	+	0.997853i	$0.479136\pi$
$938$	0	0
$939$	27374.6	0.951370
$940$	0	0
$941$	−10842.3	−0.375610	−0.187805	−	0.982206i	$-0.560137\pi$
$941$	−10842.3	−0.375610	−0.187805	+	0.982206i	$0.560137\pi$
$942$	0	0
$943$	8461.45	0.292198
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9717.58	0.333452	0.166726	−	0.986003i	$-0.446681\pi$
$947$	9717.58	0.333452	0.166726	+	0.986003i	$0.446681\pi$
$948$	0	0
$949$	5123.22	0.175244
$950$	0	0
$951$	−8689.86	−0.296307
$952$	0	0
$953$	30184.9	1.02601	0.513003	−	0.858387i	$-0.328533\pi$
$953$	30184.9	1.02601	0.513003	+	0.858387i	$0.328533\pi$
$954$	0	0
$955$	46982.1	1.59194
$956$	0	0
$957$	15041.8	0.508079
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7131.32	0.239378
$962$	0	0
$963$	15052.4	0.503692
$964$	0	0
$965$	19973.6	0.666292
$966$	0	0
$967$	−9170.72	−0.304975	−0.152487	−	0.988305i	$-0.548728\pi$
$967$	−9170.72	−0.304975	−0.152487	+	0.988305i	$0.548728\pi$
$968$	0	0
$969$	−5526.14	−0.183205
$970$	0	0
$971$	−52443.9	−1.73327	−0.866634	−	0.498944i	$-0.833721\pi$
$971$	−52443.9	−1.73327	−0.866634	+	0.498944i	$0.833721\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1714.21	0.0563062
$976$	0	0
$977$	9348.90	0.306139	0.153070	−	0.988215i	$-0.451084\pi$
$977$	9348.90	0.306139	0.153070	+	0.988215i	$0.451084\pi$
$978$	0	0
$979$	−1114.51	−0.0363841
$980$	0	0
$981$	−24770.3	−0.806173
$982$	0	0
$983$	−23736.4	−0.770167	−0.385084	−	0.922882i	$-0.625827\pi$
$983$	−23736.4	−0.770167	−0.385084	+	0.922882i	$0.625827\pi$
$984$	0	0
$985$	−32450.3	−1.04970
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5384.58	−0.173124
$990$	0	0
$991$	15639.9	0.501330	0.250665	−	0.968074i	$-0.419351\pi$
$991$	15639.9	0.501330	0.250665	+	0.968074i	$0.419351\pi$
$992$	0	0
$993$	17177.5	0.548956
$994$	0	0
$995$	−29048.2	−0.925517
$996$	0	0
$997$	17347.8	0.551064	0.275532	−	0.961292i	$-0.411146\pi$
$997$	17347.8	0.551064	0.275532	+	0.961292i	$0.411146\pi$
$998$	0	0
$999$	15889.3	0.503218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.4.a.p.1.2		2
4.3	odd	2		1568.4.a.u.1.1		2
7.6	odd	2		224.4.a.d.1.1	yes	2
21.20	even	2		2016.4.a.o.1.2		2
28.27	even	2		224.4.a.c.1.2	✓	2
56.13	odd	2		448.4.a.q.1.2		2
56.27	even	2		448.4.a.t.1.1		2
84.83	odd	2		2016.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.c.1.2	✓	2	28.27	even	2
224.4.a.d.1.1	yes	2	7.6	odd	2
448.4.a.q.1.2		2	56.13	odd	2
448.4.a.t.1.1		2	56.27	even	2
1568.4.a.p.1.2		2	1.1	even	1	trivial
1568.4.a.u.1.1		2	4.3	odd	2
2016.4.a.o.1.2		2	21.20	even	2
2016.4.a.p.1.2		2	84.83	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q+3.08276 q^{3} +9.08276 q^{5} -17.4966 q^{9} +34.1655 q^{11} -13.0828 q^{13} +28.0000 q^{15} +12.8276 q^{17} -139.745 q^{19} -21.3242 q^{23} -42.5034 q^{25} -137.172 q^{27} +142.814 q^{29} -192.152 q^{31} +105.324 q^{33} -115.834 q^{37} -40.3311 q^{39} -396.800 q^{41} +252.510 q^{43} -158.917 q^{45} -11.8208 q^{47} +39.5445 q^{51} -295.297 q^{53} +310.317 q^{55} -430.800 q^{57} -269.731 q^{59} +233.745 q^{61} -118.828 q^{65} +975.918 q^{67} -65.7374 q^{69} -870.317 q^{71} -391.600 q^{73} -131.028 q^{75} -897.573 q^{79} +49.5377 q^{81} +658.159 q^{83} +116.510 q^{85} +440.261 q^{87} -32.6210 q^{89} -592.358 q^{93} -1269.27 q^{95} -991.145 q^{97} -597.780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} + 38 q^{9} + 44 q^{11} - 14 q^{13} + 56 q^{15} - 96 q^{17} - 170 q^{19} + 152 q^{23} - 158 q^{25} - 396 q^{27} - 128 q^{29} - 68 q^{31} + 16 q^{33} - 256 q^{37} - 32 q^{39} - 88 q^{41}+ \cdots - 52 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 + 38 * q^9 + 44 * q^11 - 14 * q^13 + 56 * q^15 - 96 * q^17 - 170 * q^19 + 152 * q^23 - 158 * q^25 - 396 * q^27 - 128 * q^29 - 68 * q^31 + 16 * q^33 - 256 * q^37 - 32 * q^39 - 88 * q^41 + 724 * q^43 - 330 * q^45 + 244 * q^47 + 1028 * q^51 + 188 * q^53 + 280 * q^55 - 156 * q^57 - 138 * q^59 + 358 * q^61 - 116 * q^65 + 200 * q^67 - 1640 * q^69 - 1400 * q^71 + 628 * q^73 + 918 * q^75 + 200 * q^79 + 902 * q^81 - 618 * q^83 + 452 * q^85 + 2900 * q^87 + 908 * q^89 - 1720 * q^93 - 1176 * q^95 - 1520 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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