LMFDB - Embedded newform 3808.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3808,2,Mod(1,3808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-2,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.804272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 6x^{2} + 3x + 2 $ x^5 - 9x^3 - 6x^2 + 3*x + 2
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			$-0.783617$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.783617 q^{3} +0.768652 q^{5} -1.00000 q^{7} -2.38595 q^{9} -1.28554 q^{11} +1.08825 q^{13} -0.602329 q^{15} -1.00000 q^{17} +5.47833 q^{19} +0.783617 q^{21} +6.37379 q^{23} -4.40917 q^{25} +4.22052 q^{27} -6.89097 q^{29} -6.41522 q^{31} +1.00737 q^{33} -0.768652 q^{35} -1.90283 q^{37} -0.852773 q^{39} +1.60646 q^{41} +5.32829 q^{43} -1.83396 q^{45} +5.18542 q^{47} +1.00000 q^{49} +0.783617 q^{51} -9.60064 q^{53} -0.988133 q^{55} -4.29291 q^{57} +8.30919 q^{59} +7.74477 q^{61} +2.38595 q^{63} +0.836488 q^{65} -11.0707 q^{67} -4.99461 q^{69} +5.36995 q^{71} +9.01755 q^{73} +3.45510 q^{75} +1.28554 q^{77} +8.85277 q^{79} +3.85057 q^{81} -5.32726 q^{83} -0.768652 q^{85} +5.39987 q^{87} +4.36015 q^{89} -1.08825 q^{91} +5.02707 q^{93} +4.21093 q^{95} -13.2218 q^{97} +3.06723 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * 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
$3$	−0.783617	−0.452421	−0.226211	−	0.974078i	$-0.572634\pi$
$3$	−0.783617	−0.452421	−0.226211	+	0.974078i	$0.572634\pi$
$4$	0	0
$5$	0.768652	0.343752	0.171876	−	0.985119i	$-0.445017\pi$
$5$	0.768652	0.343752	0.171876	+	0.985119i	$0.445017\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.38595	−0.795315
$10$	0	0
$11$	−1.28554	−0.387605	−0.193802	−	0.981041i	$-0.562082\pi$
$11$	−1.28554	−0.387605	−0.193802	+	0.981041i	$0.562082\pi$
$12$	0	0
$13$	1.08825	0.301827	0.150913	−	0.988547i	$-0.451779\pi$
$13$	1.08825	0.301827	0.150913	+	0.988547i	$0.451779\pi$
$14$	0	0
$15$	−0.602329	−0.155521
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.47833	1.25682	0.628408	−	0.777884i	$-0.283707\pi$
$19$	5.47833	1.25682	0.628408	+	0.777884i	$0.283707\pi$
$20$	0	0
$21$	0.783617	0.170999
$22$	0	0
$23$	6.37379	1.32903	0.664514	−	0.747276i	$-0.268639\pi$
$23$	6.37379	1.32903	0.664514	+	0.747276i	$0.268639\pi$
$24$	0	0
$25$	−4.40917	−0.881835
$26$	0	0
$27$	4.22052	0.812239
$28$	0	0
$29$	−6.89097	−1.27962	−0.639810	−	0.768533i	$-0.720987\pi$
$29$	−6.89097	−1.27962	−0.639810	+	0.768533i	$0.720987\pi$
$30$	0	0
$31$	−6.41522	−1.15221	−0.576104	−	0.817376i	$-0.695428\pi$
$31$	−6.41522	−1.15221	−0.576104	+	0.817376i	$0.695428\pi$
$32$	0	0
$33$	1.00737	0.175361
$34$	0	0
$35$	−0.768652	−0.129926
$36$	0	0
$37$	−1.90283	−0.312824	−0.156412	−	0.987692i	$-0.549993\pi$
$37$	−1.90283	−0.312824	−0.156412	+	0.987692i	$0.549993\pi$
$38$	0	0
$39$	−0.852773	−0.136553
$40$	0	0
$41$	1.60646	0.250887	0.125444	−	0.992101i	$-0.459965\pi$
$41$	1.60646	0.250887	0.125444	+	0.992101i	$0.459965\pi$
$42$	0	0
$43$	5.32829	0.812557	0.406278	−	0.913749i	$-0.366826\pi$
$43$	5.32829	0.812557	0.406278	+	0.913749i	$0.366826\pi$
$44$	0	0
$45$	−1.83396	−0.273391
$46$	0	0
$47$	5.18542	0.756371	0.378186	−	0.925730i	$-0.376548\pi$
$47$	5.18542	0.756371	0.378186	+	0.925730i	$0.376548\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.783617	0.109728
$52$	0	0
$53$	−9.60064	−1.31875	−0.659375	−	0.751814i	$-0.729179\pi$
$53$	−9.60064	−1.31875	−0.659375	+	0.751814i	$0.729179\pi$
$54$	0	0
$55$	−0.988133	−0.133240
$56$	0	0
$57$	−4.29291	−0.568610
$58$	0	0
$59$	8.30919	1.08177	0.540883	−	0.841098i	$-0.318090\pi$
$59$	8.30919	1.08177	0.540883	+	0.841098i	$0.318090\pi$
$60$	0	0
$61$	7.74477	0.991617	0.495808	−	0.868432i	$-0.334872\pi$
$61$	7.74477	0.991617	0.495808	+	0.868432i	$0.334872\pi$
$62$	0	0
$63$	2.38595	0.300601
$64$	0	0
$65$	0.836488	0.103754
$66$	0	0
$67$	−11.0707	−1.35250	−0.676251	−	0.736671i	$-0.736397\pi$
$67$	−11.0707	−1.35250	−0.676251	+	0.736671i	$0.736397\pi$
$68$	0	0
$69$	−4.99461	−0.601280
$70$	0	0
$71$	5.36995	0.637295	0.318648	−	0.947873i	$-0.396771\pi$
$71$	5.36995	0.637295	0.318648	+	0.947873i	$0.396771\pi$
$72$	0	0
$73$	9.01755	1.05542	0.527712	−	0.849423i	$-0.323050\pi$
$73$	9.01755	1.05542	0.527712	+	0.849423i	$0.323050\pi$
$74$	0	0
$75$	3.45510	0.398961
$76$	0	0
$77$	1.28554	0.146501
$78$	0	0
$79$	8.85277	0.996015	0.498007	−	0.867173i	$-0.334065\pi$
$79$	8.85277	0.996015	0.498007	+	0.867173i	$0.334065\pi$
$80$	0	0
$81$	3.85057	0.427841
$82$	0	0
$83$	−5.32726	−0.584742	−0.292371	−	0.956305i	$-0.594444\pi$
$83$	−5.32726	−0.584742	−0.292371	+	0.956305i	$0.594444\pi$
$84$	0	0
$85$	−0.768652	−0.0833720
$86$	0	0
$87$	5.39987	0.578927
$88$	0	0
$89$	4.36015	0.462175	0.231087	−	0.972933i	$-0.425772\pi$
$89$	4.36015	0.462175	0.231087	+	0.972933i	$0.425772\pi$
$90$	0	0
$91$	−1.08825	−0.114080
$92$	0	0
$93$	5.02707	0.521283
$94$	0	0
$95$	4.21093	0.432032
$96$	0	0
$97$	−13.2218	−1.34247	−0.671234	−	0.741245i	$-0.734236\pi$
$97$	−13.2218	−1.34247	−0.671234	+	0.741245i	$0.734236\pi$
$98$	0	0
$99$	3.06723	0.308268
$100$	0	0
$101$	7.25200	0.721601	0.360800	−	0.932643i	$-0.382504\pi$
$101$	7.25200	0.721601	0.360800	+	0.932643i	$0.382504\pi$
$102$	0	0
$103$	6.11640	0.602667	0.301334	−	0.953519i	$-0.402568\pi$
$103$	6.11640	0.602667	0.301334	+	0.953519i	$0.402568\pi$
$104$	0	0
$105$	0.602329	0.0587813
$106$	0	0
$107$	4.47898	0.432999	0.216500	−	0.976283i	$-0.430536\pi$
$107$	4.47898	0.432999	0.216500	+	0.976283i	$0.430536\pi$
$108$	0	0
$109$	18.7923	1.79997	0.899986	−	0.435919i	$-0.143577\pi$
$109$	18.7923	1.79997	0.899986	+	0.435919i	$0.143577\pi$
$110$	0	0
$111$	1.49109	0.141528
$112$	0	0
$113$	17.8094	1.67537	0.837685	−	0.546153i	$-0.183908\pi$
$113$	17.8094	1.67537	0.837685	+	0.546153i	$0.183908\pi$
$114$	0	0
$115$	4.89923	0.456856
$116$	0	0
$117$	−2.59651	−0.240048
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−9.34739	−0.849763
$122$	0	0
$123$	−1.25885	−0.113507
$124$	0	0
$125$	−7.23238	−0.646884
$126$	0	0
$127$	12.2550	1.08746	0.543728	−	0.839262i	$-0.317012\pi$
$127$	12.2550	1.08746	0.543728	+	0.839262i	$0.317012\pi$
$128$	0	0
$129$	−4.17534	−0.367618
$130$	0	0
$131$	−7.87643	−0.688167	−0.344083	−	0.938939i	$-0.611810\pi$
$131$	−7.87643	−0.688167	−0.344083	+	0.938939i	$0.611810\pi$
$132$	0	0
$133$	−5.47833	−0.475031
$134$	0	0
$135$	3.24411	0.279208
$136$	0	0
$137$	17.8546	1.52542	0.762711	−	0.646739i	$-0.223868\pi$
$137$	17.8546	1.52542	0.762711	+	0.646739i	$0.223868\pi$
$138$	0	0
$139$	12.1290	1.02877	0.514386	−	0.857559i	$-0.328020\pi$
$139$	12.1290	1.02877	0.514386	+	0.857559i	$0.328020\pi$
$140$	0	0
$141$	−4.06338	−0.342198
$142$	0	0
$143$	−1.39899	−0.116990
$144$	0	0
$145$	−5.29676	−0.439872
$146$	0	0
$147$	−0.783617	−0.0646316
$148$	0	0
$149$	−8.18106	−0.670219	−0.335109	−	0.942179i	$-0.608773\pi$
$149$	−8.18106	−0.670219	−0.335109	+	0.942179i	$0.608773\pi$
$150$	0	0
$151$	20.6449	1.68006	0.840028	−	0.542542i	$-0.182538\pi$
$151$	20.6449	1.68006	0.840028	+	0.542542i	$0.182538\pi$
$152$	0	0
$153$	2.38595	0.192892
$154$	0	0
$155$	−4.93108	−0.396074
$156$	0	0
$157$	−22.1459	−1.76744	−0.883718	−	0.468019i	$-0.844968\pi$
$157$	−22.1459	−1.76744	−0.883718	+	0.468019i	$0.844968\pi$
$158$	0	0
$159$	7.52322	0.596630
$160$	0	0
$161$	−6.37379	−0.502325
$162$	0	0
$163$	−5.12025	−0.401049	−0.200525	−	0.979689i	$-0.564265\pi$
$163$	−5.12025	−0.401049	−0.200525	+	0.979689i	$0.564265\pi$
$164$	0	0
$165$	0.774317	0.0602805
$166$	0	0
$167$	18.3330	1.41865	0.709326	−	0.704881i	$-0.248999\pi$
$167$	18.3330	1.41865	0.709326	+	0.704881i	$0.248999\pi$
$168$	0	0
$169$	−11.8157	−0.908900
$170$	0	0
$171$	−13.0710	−0.999564
$172$	0	0
$173$	6.10173	0.463906	0.231953	−	0.972727i	$-0.425489\pi$
$173$	6.10173	0.463906	0.231953	+	0.972727i	$0.425489\pi$
$174$	0	0
$175$	4.40917	0.333302
$176$	0	0
$177$	−6.51122	−0.489413
$178$	0	0
$179$	4.43019	0.331128	0.165564	−	0.986199i	$-0.447056\pi$
$179$	4.43019	0.331128	0.165564	+	0.986199i	$0.447056\pi$
$180$	0	0
$181$	12.6622	0.941174	0.470587	−	0.882354i	$-0.344042\pi$
$181$	12.6622	0.941174	0.470587	+	0.882354i	$0.344042\pi$
$182$	0	0
$183$	−6.06893	−0.448628
$184$	0	0
$185$	−1.46262	−0.107534
$186$	0	0
$187$	1.28554	0.0940080
$188$	0	0
$189$	−4.22052	−0.306997
$190$	0	0
$191$	−7.62969	−0.552065	−0.276032	−	0.961148i	$-0.589020\pi$
$191$	−7.62969	−0.552065	−0.276032	+	0.961148i	$0.589020\pi$
$192$	0	0
$193$	−7.51145	−0.540686	−0.270343	−	0.962764i	$-0.587137\pi$
$193$	−7.51145	−0.540686	−0.270343	+	0.962764i	$0.587137\pi$
$194$	0	0
$195$	−0.655486	−0.0469403
$196$	0	0
$197$	−6.86399	−0.489039	−0.244519	−	0.969644i	$-0.578630\pi$
$197$	−6.86399	−0.489039	−0.244519	+	0.969644i	$0.578630\pi$
$198$	0	0
$199$	−6.37948	−0.452229	−0.226115	−	0.974101i	$-0.572602\pi$
$199$	−6.37948	−0.452229	−0.226115	+	0.974101i	$0.572602\pi$
$200$	0	0
$201$	8.67519	0.611901
$202$	0	0
$203$	6.89097	0.483651
$204$	0	0
$205$	1.23481	0.0862428
$206$	0	0
$207$	−15.2075	−1.05700
$208$	0	0
$209$	−7.04261	−0.487148
$210$	0	0
$211$	−18.4176	−1.26792	−0.633959	−	0.773366i	$-0.718571\pi$
$211$	−18.4176	−1.26792	−0.633959	+	0.773366i	$0.718571\pi$
$212$	0	0
$213$	−4.20798	−0.288326
$214$	0	0
$215$	4.09560	0.279318
$216$	0	0
$217$	6.41522	0.435494
$218$	0	0
$219$	−7.06630	−0.477496
$220$	0	0
$221$	−1.08825	−0.0732038
$222$	0	0
$223$	−7.71469	−0.516614	−0.258307	−	0.966063i	$-0.583165\pi$
$223$	−7.71469	−0.516614	−0.258307	+	0.966063i	$0.583165\pi$
$224$	0	0
$225$	10.5200	0.701336
$226$	0	0
$227$	25.5319	1.69461	0.847305	−	0.531107i	$-0.178224\pi$
$227$	25.5319	1.69461	0.847305	+	0.531107i	$0.178224\pi$
$228$	0	0
$229$	28.0484	1.85349	0.926746	−	0.375687i	$-0.122593\pi$
$229$	28.0484	1.85349	0.926746	+	0.375687i	$0.122593\pi$
$230$	0	0
$231$	−1.00737	−0.0662801
$232$	0	0
$233$	3.67002	0.240431	0.120216	−	0.992748i	$-0.461641\pi$
$233$	3.67002	0.240431	0.120216	+	0.992748i	$0.461641\pi$
$234$	0	0
$235$	3.98578	0.260004
$236$	0	0
$237$	−6.93718	−0.450618
$238$	0	0
$239$	18.4710	1.19479	0.597396	−	0.801947i	$-0.296202\pi$
$239$	18.4710	1.19479	0.597396	+	0.801947i	$0.296202\pi$
$240$	0	0
$241$	21.4990	1.38487	0.692435	−	0.721480i	$-0.256538\pi$
$241$	21.4990	1.38487	0.692435	+	0.721480i	$0.256538\pi$
$242$	0	0
$243$	−15.6789	−1.00580
$244$	0	0
$245$	0.768652	0.0491074
$246$	0	0
$247$	5.96181	0.379341
$248$	0	0
$249$	4.17453	0.264550
$250$	0	0
$251$	−18.8219	−1.18803	−0.594013	−	0.804455i	$-0.702457\pi$
$251$	−18.8219	−1.18803	−0.594013	+	0.804455i	$0.702457\pi$
$252$	0	0
$253$	−8.19376	−0.515137
$254$	0	0
$255$	0.602329	0.0377193
$256$	0	0
$257$	3.00737	0.187595	0.0937973	−	0.995591i	$-0.470099\pi$
$257$	3.00737	0.187595	0.0937973	+	0.995591i	$0.470099\pi$
$258$	0	0
$259$	1.90283	0.118236
$260$	0	0
$261$	16.4415	1.01770
$262$	0	0
$263$	20.5885	1.26954	0.634769	−	0.772702i	$-0.281095\pi$
$263$	20.5885	1.26954	0.634769	+	0.772702i	$0.281095\pi$
$264$	0	0
$265$	−7.37956	−0.453323
$266$	0	0
$267$	−3.41668	−0.209098
$268$	0	0
$269$	−1.96181	−0.119613	−0.0598067	−	0.998210i	$-0.519048\pi$
$269$	−1.96181	−0.119613	−0.0598067	+	0.998210i	$0.519048\pi$
$270$	0	0
$271$	21.2546	1.29113	0.645564	−	0.763706i	$-0.276623\pi$
$271$	21.2546	1.29113	0.645564	+	0.763706i	$0.276623\pi$
$272$	0	0
$273$	0.852773	0.0516122
$274$	0	0
$275$	5.66817	0.341803
$276$	0	0
$277$	21.7614	1.30752	0.653759	−	0.756703i	$-0.273191\pi$
$277$	21.7614	1.30752	0.653759	+	0.756703i	$0.273191\pi$
$278$	0	0
$279$	15.3064	0.916368
$280$	0	0
$281$	9.77095	0.582886	0.291443	−	0.956588i	$-0.405865\pi$
$281$	9.77095	0.582886	0.291443	+	0.956588i	$0.405865\pi$
$282$	0	0
$283$	−5.40135	−0.321077	−0.160538	−	0.987030i	$-0.551323\pi$
$283$	−5.40135	−0.321077	−0.160538	+	0.987030i	$0.551323\pi$
$284$	0	0
$285$	−3.29975	−0.195461
$286$	0	0
$287$	−1.60646	−0.0948264
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	10.3608	0.607361
$292$	0	0
$293$	−3.06546	−0.179086	−0.0895431	−	0.995983i	$-0.528541\pi$
$293$	−3.06546	−0.179086	−0.0895431	+	0.995983i	$0.528541\pi$
$294$	0	0
$295$	6.38688	0.371859
$296$	0	0
$297$	−5.42564	−0.314828
$298$	0	0
$299$	6.93630	0.401136
$300$	0	0
$301$	−5.32829	−0.307118
$302$	0	0
$303$	−5.68278	−0.326467
$304$	0	0
$305$	5.95304	0.340870
$306$	0	0
$307$	−26.7760	−1.52819	−0.764093	−	0.645107i	$-0.776813\pi$
$307$	−26.7760	−1.52819	−0.764093	+	0.645107i	$0.776813\pi$
$308$	0	0
$309$	−4.79292	−0.272659
$310$	0	0
$311$	23.6065	1.33860	0.669300	−	0.742992i	$-0.266594\pi$
$311$	23.6065	1.33860	0.669300	+	0.742992i	$0.266594\pi$
$312$	0	0
$313$	0.668533	0.0377877	0.0188938	−	0.999821i	$-0.493986\pi$
$313$	0.668533	0.0377877	0.0188938	+	0.999821i	$0.493986\pi$
$314$	0	0
$315$	1.83396	0.103332
$316$	0	0
$317$	−10.5586	−0.593028	−0.296514	−	0.955029i	$-0.595824\pi$
$317$	−10.5586	−0.593028	−0.296514	+	0.955029i	$0.595824\pi$
$318$	0	0
$319$	8.85861	0.495987
$320$	0	0
$321$	−3.50980	−0.195898
$322$	0	0
$323$	−5.47833	−0.304822
$324$	0	0
$325$	−4.79830	−0.266162
$326$	0	0
$327$	−14.7259	−0.814345
$328$	0	0
$329$	−5.18542	−0.285881
$330$	0	0
$331$	12.0313	0.661301	0.330650	−	0.943753i	$-0.392732\pi$
$331$	12.0313	0.661301	0.330650	+	0.943753i	$0.392732\pi$
$332$	0	0
$333$	4.54005	0.248793
$334$	0	0
$335$	−8.50953	−0.464925
$336$	0	0
$337$	−3.42628	−0.186641	−0.0933207	−	0.995636i	$-0.529748\pi$
$337$	−3.42628	−0.186641	−0.0933207	+	0.995636i	$0.529748\pi$
$338$	0	0
$339$	−13.9558	−0.757973
$340$	0	0
$341$	8.24702	0.446601
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.83912	−0.206691
$346$	0	0
$347$	4.40931	0.236704	0.118352	−	0.992972i	$-0.462239\pi$
$347$	4.40931	0.236704	0.118352	+	0.992972i	$0.462239\pi$
$348$	0	0
$349$	−14.3758	−0.769517	−0.384759	−	0.923017i	$-0.625715\pi$
$349$	−14.3758	−0.769517	−0.384759	+	0.923017i	$0.625715\pi$
$350$	0	0
$351$	4.59299	0.245156
$352$	0	0
$353$	−12.1664	−0.647551	−0.323776	−	0.946134i	$-0.604952\pi$
$353$	−12.1664	−0.647551	−0.323776	+	0.946134i	$0.604952\pi$
$354$	0	0
$355$	4.12762	0.219071
$356$	0	0
$357$	−0.783617	−0.0414734
$358$	0	0
$359$	19.1027	1.00820	0.504101	−	0.863645i	$-0.331824\pi$
$359$	19.1027	1.00820	0.504101	+	0.863645i	$0.331824\pi$
$360$	0	0
$361$	11.0121	0.579584
$362$	0	0
$363$	7.32477	0.384451
$364$	0	0
$365$	6.93136	0.362804
$366$	0	0
$367$	4.12275	0.215206	0.107603	−	0.994194i	$-0.465682\pi$
$367$	4.12275	0.215206	0.107603	+	0.994194i	$0.465682\pi$
$368$	0	0
$369$	−3.83293	−0.199534
$370$	0	0
$371$	9.60064	0.498441
$372$	0	0
$373$	4.88585	0.252980	0.126490	−	0.991968i	$-0.459629\pi$
$373$	4.88585	0.252980	0.126490	+	0.991968i	$0.459629\pi$
$374$	0	0
$375$	5.66741	0.292664
$376$	0	0
$377$	−7.49911	−0.386224
$378$	0	0
$379$	9.40040	0.482866	0.241433	−	0.970417i	$-0.422383\pi$
$379$	9.40040	0.482866	0.241433	+	0.970417i	$0.422383\pi$
$380$	0	0
$381$	−9.60322	−0.491988
$382$	0	0
$383$	0.211502	0.0108073	0.00540363	−	0.999985i	$-0.498280\pi$
$383$	0.211502	0.0108073	0.00540363	+	0.999985i	$0.498280\pi$
$384$	0	0
$385$	0.988133	0.0503599
$386$	0	0
$387$	−12.7130	−0.646239
$388$	0	0
$389$	21.1297	1.07132	0.535659	−	0.844435i	$-0.320063\pi$
$389$	21.1297	1.07132	0.535659	+	0.844435i	$0.320063\pi$
$390$	0	0
$391$	−6.37379	−0.322337
$392$	0	0
$393$	6.17210	0.311341
$394$	0	0
$395$	6.80470	0.342382
$396$	0	0
$397$	4.29826	0.215724	0.107862	−	0.994166i	$-0.465600\pi$
$397$	4.29826	0.215724	0.107862	+	0.994166i	$0.465600\pi$
$398$	0	0
$399$	4.29291	0.214914
$400$	0	0
$401$	−3.49222	−0.174393	−0.0871965	−	0.996191i	$-0.527791\pi$
$401$	−3.49222	−0.174393	−0.0871965	+	0.996191i	$0.527791\pi$
$402$	0	0
$403$	−6.98138	−0.347767
$404$	0	0
$405$	2.95975	0.147071
$406$	0	0
$407$	2.44617	0.121252
$408$	0	0
$409$	−36.4671	−1.80318	−0.901592	−	0.432587i	$-0.857601\pi$
$409$	−36.4671	−1.80318	−0.901592	+	0.432587i	$0.857601\pi$
$410$	0	0
$411$	−13.9912	−0.690133
$412$	0	0
$413$	−8.30919	−0.408869
$414$	0	0
$415$	−4.09481	−0.201006
$416$	0	0
$417$	−9.50451	−0.465438
$418$	0	0
$419$	−27.2865	−1.33303	−0.666516	−	0.745490i	$-0.732215\pi$
$419$	−27.2865	−1.33303	−0.666516	+	0.745490i	$0.732215\pi$
$420$	0	0
$421$	−29.3003	−1.42801	−0.714006	−	0.700140i	$-0.753121\pi$
$421$	−29.3003	−1.42801	−0.714006	+	0.700140i	$0.753121\pi$
$422$	0	0
$423$	−12.3721	−0.601553
$424$	0	0
$425$	4.40917	0.213876
$426$	0	0
$427$	−7.74477	−0.374796
$428$	0	0
$429$	1.09627	0.0529286
$430$	0	0
$431$	−29.9041	−1.44043	−0.720214	−	0.693751i	$-0.755957\pi$
$431$	−29.9041	−1.44043	−0.720214	+	0.693751i	$0.755957\pi$
$432$	0	0
$433$	35.4200	1.70218	0.851089	−	0.525021i	$-0.175943\pi$
$433$	35.4200	1.70218	0.851089	+	0.525021i	$0.175943\pi$
$434$	0	0
$435$	4.15063	0.199007
$436$	0	0
$437$	34.9177	1.67034
$438$	0	0
$439$	24.8448	1.18578	0.592889	−	0.805284i	$-0.297987\pi$
$439$	24.8448	1.18578	0.592889	+	0.805284i	$0.297987\pi$
$440$	0	0
$441$	−2.38595	−0.113616
$442$	0	0
$443$	33.2096	1.57783	0.788917	−	0.614500i	$-0.210642\pi$
$443$	33.2096	1.57783	0.788917	+	0.614500i	$0.210642\pi$
$444$	0	0
$445$	3.35144	0.158873
$446$	0	0
$447$	6.41082	0.303221
$448$	0	0
$449$	−1.92122	−0.0906678	−0.0453339	−	0.998972i	$-0.514435\pi$
$449$	−1.92122	−0.0906678	−0.0453339	+	0.998972i	$0.514435\pi$
$450$	0	0
$451$	−2.06517	−0.0972450
$452$	0	0
$453$	−16.1777	−0.760093
$454$	0	0
$455$	−0.836488	−0.0392152
$456$	0	0
$457$	25.0347	1.17108	0.585538	−	0.810645i	$-0.300883\pi$
$457$	25.0347	1.17108	0.585538	+	0.810645i	$0.300883\pi$
$458$	0	0
$459$	−4.22052	−0.196997
$460$	0	0
$461$	−19.4920	−0.907832	−0.453916	−	0.891045i	$-0.649973\pi$
$461$	−19.4920	−0.907832	−0.453916	+	0.891045i	$0.649973\pi$
$462$	0	0
$463$	35.0356	1.62824	0.814122	−	0.580694i	$-0.197219\pi$
$463$	35.0356	1.62824	0.814122	+	0.580694i	$0.197219\pi$
$464$	0	0
$465$	3.86407	0.179192
$466$	0	0
$467$	1.25691	0.0581630	0.0290815	−	0.999577i	$-0.490742\pi$
$467$	1.25691	0.0581630	0.0290815	+	0.999577i	$0.490742\pi$
$468$	0	0
$469$	11.0707	0.511198
$470$	0	0
$471$	17.3539	0.799626
$472$	0	0
$473$	−6.84973	−0.314951
$474$	0	0
$475$	−24.1549	−1.10830
$476$	0	0
$477$	22.9066	1.04882
$478$	0	0
$479$	11.8058	0.539419	0.269709	−	0.962942i	$-0.413072\pi$
$479$	11.8058	0.539419	0.269709	+	0.962942i	$0.413072\pi$
$480$	0	0
$481$	−2.07076	−0.0944186
$482$	0	0
$483$	4.99461	0.227263
$484$	0	0
$485$	−10.1630	−0.461476
$486$	0	0
$487$	−19.4622	−0.881916	−0.440958	−	0.897528i	$-0.645361\pi$
$487$	−19.4622	−0.881916	−0.440958	+	0.897528i	$0.645361\pi$
$488$	0	0
$489$	4.01231	0.181443
$490$	0	0
$491$	17.3951	0.785030	0.392515	−	0.919746i	$-0.371605\pi$
$491$	17.3951	0.785030	0.392515	+	0.919746i	$0.371605\pi$
$492$	0	0
$493$	6.89097	0.310353
$494$	0	0
$495$	2.35763	0.105968
$496$	0	0
$497$	−5.36995	−0.240875
$498$	0	0
$499$	−6.33937	−0.283789	−0.141895	−	0.989882i	$-0.545319\pi$
$499$	−6.33937	−0.283789	−0.141895	+	0.989882i	$0.545319\pi$
$500$	0	0
$501$	−14.3661	−0.641828
$502$	0	0
$503$	−5.96727	−0.266067	−0.133034	−	0.991112i	$-0.542472\pi$
$503$	−5.96727	−0.266067	−0.133034	+	0.991112i	$0.542472\pi$
$504$	0	0
$505$	5.57426	0.248051
$506$	0	0
$507$	9.25898	0.411206
$508$	0	0
$509$	4.31074	0.191070	0.0955351	−	0.995426i	$-0.469544\pi$
$509$	4.31074	0.191070	0.0955351	+	0.995426i	$0.469544\pi$
$510$	0	0
$511$	−9.01755	−0.398913
$512$	0	0
$513$	23.1214	1.02083
$514$	0	0
$515$	4.70139	0.207168
$516$	0	0
$517$	−6.66606	−0.293173
$518$	0	0
$519$	−4.78141	−0.209881
$520$	0	0
$521$	−29.3337	−1.28513	−0.642567	−	0.766230i	$-0.722130\pi$
$521$	−29.3337	−1.28513	−0.642567	+	0.766230i	$0.722130\pi$
$522$	0	0
$523$	−19.9821	−0.873758	−0.436879	−	0.899520i	$-0.643916\pi$
$523$	−19.9821	−0.873758	−0.436879	+	0.899520i	$0.643916\pi$
$524$	0	0
$525$	−3.45510	−0.150793
$526$	0	0
$527$	6.41522	0.279452
$528$	0	0
$529$	17.6252	0.766314
$530$	0	0
$531$	−19.8253	−0.860344
$532$	0	0
$533$	1.74824	0.0757245
$534$	0	0
$535$	3.44278	0.148844
$536$	0	0
$537$	−3.47157	−0.149809
$538$	0	0
$539$	−1.28554	−0.0553721
$540$	0	0
$541$	13.0394	0.560606	0.280303	−	0.959912i	$-0.409565\pi$
$541$	13.0394	0.560606	0.280303	+	0.959912i	$0.409565\pi$
$542$	0	0
$543$	−9.92231	−0.425807
$544$	0	0
$545$	14.4447	0.618743
$546$	0	0
$547$	−28.2140	−1.20635	−0.603173	−	0.797611i	$-0.706097\pi$
$547$	−28.2140	−1.20635	−0.603173	+	0.797611i	$0.706097\pi$
$548$	0	0
$549$	−18.4786	−0.788648
$550$	0	0
$551$	−37.7510	−1.60825
$552$	0	0
$553$	−8.85277	−0.376458
$554$	0	0
$555$	1.14613	0.0486505
$556$	0	0
$557$	18.3281	0.776585	0.388292	−	0.921536i	$-0.373065\pi$
$557$	18.3281	0.776585	0.388292	+	0.921536i	$0.373065\pi$
$558$	0	0
$559$	5.79853	0.245252
$560$	0	0
$561$	−1.00737	−0.0425312
$562$	0	0
$563$	−42.9382	−1.80963	−0.904814	−	0.425807i	$-0.859990\pi$
$563$	−42.9382	−1.80963	−0.904814	+	0.425807i	$0.859990\pi$
$564$	0	0
$565$	13.6893	0.575912
$566$	0	0
$567$	−3.85057	−0.161709
$568$	0	0
$569$	−37.8151	−1.58529	−0.792646	−	0.609682i	$-0.791297\pi$
$569$	−37.8151	−1.58529	−0.792646	+	0.609682i	$0.791297\pi$
$570$	0	0
$571$	14.9518	0.625715	0.312857	−	0.949800i	$-0.398714\pi$
$571$	14.9518	0.625715	0.312857	+	0.949800i	$0.398714\pi$
$572$	0	0
$573$	5.97875	0.249766
$574$	0	0
$575$	−28.1032	−1.17198
$576$	0	0
$577$	−25.3420	−1.05500	−0.527501	−	0.849555i	$-0.676871\pi$
$577$	−25.3420	−1.05500	−0.527501	+	0.849555i	$0.676871\pi$
$578$	0	0
$579$	5.88610	0.244618
$580$	0	0
$581$	5.32726	0.221012
$582$	0	0
$583$	12.3420	0.511154
$584$	0	0
$585$	−1.99581	−0.0825168
$586$	0	0
$587$	31.1377	1.28519	0.642594	−	0.766207i	$-0.277858\pi$
$587$	31.1377	1.28519	0.642594	+	0.766207i	$0.277858\pi$
$588$	0	0
$589$	−35.1447	−1.44811
$590$	0	0
$591$	5.37874	0.221252
$592$	0	0
$593$	−18.3651	−0.754163	−0.377082	−	0.926180i	$-0.623072\pi$
$593$	−18.3651	−0.754163	−0.377082	+	0.926180i	$0.623072\pi$
$594$	0	0
$595$	0.768652	0.0315117
$596$	0	0
$597$	4.99906	0.204598
$598$	0	0
$599$	−34.8865	−1.42542	−0.712712	−	0.701456i	$-0.752534\pi$
$599$	−34.8865	−1.42542	−0.712712	+	0.701456i	$0.752534\pi$
$600$	0	0
$601$	−10.1224	−0.412899	−0.206450	−	0.978457i	$-0.566191\pi$
$601$	−10.1224	−0.412899	−0.206450	+	0.978457i	$0.566191\pi$
$602$	0	0
$603$	26.4141	1.07567
$604$	0	0
$605$	−7.18489	−0.292107
$606$	0	0
$607$	10.0180	0.406618	0.203309	−	0.979115i	$-0.434830\pi$
$607$	10.0180	0.406618	0.203309	+	0.979115i	$0.434830\pi$
$608$	0	0
$609$	−5.39987	−0.218814
$610$	0	0
$611$	5.64305	0.228293
$612$	0	0
$613$	−3.16466	−0.127820	−0.0639098	−	0.997956i	$-0.520357\pi$
$613$	−3.16466	−0.127820	−0.0639098	+	0.997956i	$0.520357\pi$
$614$	0	0
$615$	−0.967617	−0.0390181
$616$	0	0
$617$	−10.4545	−0.420884	−0.210442	−	0.977606i	$-0.567490\pi$
$617$	−10.4545	−0.420884	−0.210442	+	0.977606i	$0.567490\pi$
$618$	0	0
$619$	−37.9814	−1.52660	−0.763302	−	0.646042i	$-0.776423\pi$
$619$	−37.9814	−1.52660	−0.763302	+	0.646042i	$0.776423\pi$
$620$	0	0
$621$	26.9007	1.07949
$622$	0	0
$623$	−4.36015	−0.174686
$624$	0	0
$625$	16.4867	0.659467
$626$	0	0
$627$	5.51871	0.220396
$628$	0	0
$629$	1.90283	0.0758709
$630$	0	0
$631$	−23.2117	−0.924042	−0.462021	−	0.886869i	$-0.652875\pi$
$631$	−23.2117	−0.924042	−0.462021	+	0.886869i	$0.652875\pi$
$632$	0	0
$633$	14.4323	0.573633
$634$	0	0
$635$	9.41984	0.373815
$636$	0	0
$637$	1.08825	0.0431181
$638$	0	0
$639$	−12.8124	−0.506851
$640$	0	0
$641$	28.6658	1.13223	0.566116	−	0.824326i	$-0.308445\pi$
$641$	28.6658	1.13223	0.566116	+	0.824326i	$0.308445\pi$
$642$	0	0
$643$	9.05125	0.356946	0.178473	−	0.983945i	$-0.442884\pi$
$643$	9.05125	0.356946	0.178473	+	0.983945i	$0.442884\pi$
$644$	0	0
$645$	−3.20938	−0.126369
$646$	0	0
$647$	17.6468	0.693768	0.346884	−	0.937908i	$-0.387240\pi$
$647$	17.6468	0.693768	0.346884	+	0.937908i	$0.387240\pi$
$648$	0	0
$649$	−10.6818	−0.419297
$650$	0	0
$651$	−5.02707	−0.197027
$652$	0	0
$653$	18.7515	0.733804	0.366902	−	0.930260i	$-0.380418\pi$
$653$	18.7515	0.733804	0.366902	+	0.930260i	$0.380418\pi$
$654$	0	0
$655$	−6.05423	−0.236558
$656$	0	0
$657$	−21.5154	−0.839395
$658$	0	0
$659$	−26.0064	−1.01307	−0.506533	−	0.862220i	$-0.669073\pi$
$659$	−26.0064	−1.01307	−0.506533	+	0.862220i	$0.669073\pi$
$660$	0	0
$661$	−23.8933	−0.929340	−0.464670	−	0.885484i	$-0.653827\pi$
$661$	−23.8933	−0.929340	−0.464670	+	0.885484i	$0.653827\pi$
$662$	0	0
$663$	0.852773	0.0331190
$664$	0	0
$665$	−4.21093	−0.163293
$666$	0	0
$667$	−43.9216	−1.70065
$668$	0	0
$669$	6.04536	0.233727
$670$	0	0
$671$	−9.95621	−0.384355
$672$	0	0
$673$	14.2500	0.549296	0.274648	−	0.961545i	$-0.411439\pi$
$673$	14.2500	0.549296	0.274648	+	0.961545i	$0.411439\pi$
$674$	0	0
$675$	−18.6090	−0.716260
$676$	0	0
$677$	25.9475	0.997245	0.498622	−	0.866819i	$-0.333839\pi$
$677$	25.9475	0.997245	0.498622	+	0.866819i	$0.333839\pi$
$678$	0	0
$679$	13.2218	0.507405
$680$	0	0
$681$	−20.0072	−0.766677
$682$	0	0
$683$	−44.7301	−1.71155	−0.855775	−	0.517348i	$-0.826919\pi$
$683$	−44.7301	−1.71155	−0.855775	+	0.517348i	$0.826919\pi$
$684$	0	0
$685$	13.7240	0.524367
$686$	0	0
$687$	−21.9792	−0.838559
$688$	0	0
$689$	−10.4479	−0.398034
$690$	0	0
$691$	−41.2568	−1.56948	−0.784741	−	0.619824i	$-0.787204\pi$
$691$	−41.2568	−1.56948	−0.784741	+	0.619824i	$0.787204\pi$
$692$	0	0
$693$	−3.06723	−0.116514
$694$	0	0
$695$	9.32301	0.353642
$696$	0	0
$697$	−1.60646	−0.0608490
$698$	0	0
$699$	−2.87589	−0.108776
$700$	0	0
$701$	−27.9208	−1.05455	−0.527277	−	0.849693i	$-0.676787\pi$
$701$	−27.9208	−1.05455	−0.527277	+	0.849693i	$0.676787\pi$
$702$	0	0
$703$	−10.4243	−0.393162
$704$	0	0
$705$	−3.12333	−0.117631
$706$	0	0
$707$	−7.25200	−0.272739
$708$	0	0
$709$	51.4457	1.93208	0.966042	−	0.258385i	$-0.0831902\pi$
$709$	51.4457	1.93208	0.966042	+	0.258385i	$0.0831902\pi$
$710$	0	0
$711$	−21.1222	−0.792146
$712$	0	0
$713$	−40.8893	−1.53132
$714$	0	0
$715$	−1.07534	−0.0402154
$716$	0	0
$717$	−14.4742	−0.540549
$718$	0	0
$719$	9.67403	0.360780	0.180390	−	0.983595i	$-0.442264\pi$
$719$	9.67403	0.360780	0.180390	+	0.983595i	$0.442264\pi$
$720$	0	0
$721$	−6.11640	−0.227787
$722$	0	0
$723$	−16.8470	−0.626545
$724$	0	0
$725$	30.3835	1.12841
$726$	0	0
$727$	−12.7398	−0.472495	−0.236247	−	0.971693i	$-0.575918\pi$
$727$	−12.7398	−0.472495	−0.236247	+	0.971693i	$0.575918\pi$
$728$	0	0
$729$	0.734549	0.0272055
$730$	0	0
$731$	−5.32829	−0.197074
$732$	0	0
$733$	22.2798	0.822924	0.411462	−	0.911427i	$-0.365018\pi$
$733$	22.2798	0.822924	0.411462	+	0.911427i	$0.365018\pi$
$734$	0	0
$735$	−0.602329	−0.0222172
$736$	0	0
$737$	14.2318	0.524236
$738$	0	0
$739$	−5.48049	−0.201603	−0.100802	−	0.994907i	$-0.532141\pi$
$739$	−5.48049	−0.201603	−0.100802	+	0.994907i	$0.532141\pi$
$740$	0	0
$741$	−4.67177	−0.171622
$742$	0	0
$743$	45.2576	1.66034	0.830170	−	0.557510i	$-0.188243\pi$
$743$	45.2576	1.66034	0.830170	+	0.557510i	$0.188243\pi$
$744$	0	0
$745$	−6.28839	−0.230389
$746$	0	0
$747$	12.7105	0.465054
$748$	0	0
$749$	−4.47898	−0.163658
$750$	0	0
$751$	−53.9692	−1.96936	−0.984682	−	0.174359i	$-0.944215\pi$
$751$	−53.9692	−1.96936	−0.984682	+	0.174359i	$0.944215\pi$
$752$	0	0
$753$	14.7491	0.537488
$754$	0	0
$755$	15.8687	0.577523
$756$	0	0
$757$	−11.8707	−0.431447	−0.215724	−	0.976454i	$-0.569211\pi$
$757$	−11.8707	−0.431447	−0.215724	+	0.976454i	$0.569211\pi$
$758$	0	0
$759$	6.42077	0.233059
$760$	0	0
$761$	−26.8278	−0.972507	−0.486254	−	0.873818i	$-0.661637\pi$
$761$	−26.8278	−0.972507	−0.486254	+	0.873818i	$0.661637\pi$
$762$	0	0
$763$	−18.7923	−0.680325
$764$	0	0
$765$	1.83396	0.0663070
$766$	0	0
$767$	9.04250	0.326506
$768$	0	0
$769$	−33.0913	−1.19330	−0.596652	−	0.802500i	$-0.703503\pi$
$769$	−33.0913	−1.19330	−0.596652	+	0.802500i	$0.703503\pi$
$770$	0	0
$771$	−2.35662	−0.0848718
$772$	0	0
$773$	−38.5501	−1.38655	−0.693275	−	0.720673i	$-0.743833\pi$
$773$	−38.5501	−1.38655	−0.693275	+	0.720673i	$0.743833\pi$
$774$	0	0
$775$	28.2858	1.01606
$776$	0	0
$777$	−1.49109	−0.0534926
$778$	0	0
$779$	8.80072	0.315319
$780$	0	0
$781$	−6.90328	−0.247019
$782$	0	0
$783$	−29.0834	−1.03936
$784$	0	0
$785$	−17.0225	−0.607559
$786$	0	0
$787$	−9.60159	−0.342260	−0.171130	−	0.985248i	$-0.554742\pi$
$787$	−9.60159	−0.342260	−0.171130	+	0.985248i	$0.554742\pi$
$788$	0	0
$789$	−16.1335	−0.574366
$790$	0	0
$791$	−17.8094	−0.633231
$792$	0	0
$793$	8.42827	0.299297
$794$	0	0
$795$	5.78274	0.205093
$796$	0	0
$797$	29.2539	1.03623	0.518113	−	0.855312i	$-0.326635\pi$
$797$	29.2539	1.03623	0.518113	+	0.855312i	$0.326635\pi$
$798$	0	0
$799$	−5.18542	−0.183447
$800$	0	0
$801$	−10.4031	−0.367575
$802$	0	0
$803$	−11.5924	−0.409088
$804$	0	0
$805$	−4.89923	−0.172675
$806$	0	0
$807$	1.53730	0.0541157
$808$	0	0
$809$	37.8777	1.33171	0.665855	−	0.746082i	$-0.268067\pi$
$809$	37.8777	1.33171	0.665855	+	0.746082i	$0.268067\pi$
$810$	0	0
$811$	−16.2827	−0.571764	−0.285882	−	0.958265i	$-0.592287\pi$
$811$	−16.2827	−0.571764	−0.285882	+	0.958265i	$0.592287\pi$
$812$	0	0
$813$	−16.6555	−0.584133
$814$	0	0
$815$	−3.93569	−0.137861
$816$	0	0
$817$	29.1901	1.02123
$818$	0	0
$819$	2.59651	0.0907294
$820$	0	0
$821$	−11.2464	−0.392501	−0.196251	−	0.980554i	$-0.562877\pi$
$821$	−11.2464	−0.392501	−0.196251	+	0.980554i	$0.562877\pi$
$822$	0	0
$823$	1.95608	0.0681845	0.0340923	−	0.999419i	$-0.489146\pi$
$823$	1.95608	0.0681845	0.0340923	+	0.999419i	$0.489146\pi$
$824$	0	0
$825$	−4.44167	−0.154639
$826$	0	0
$827$	39.9767	1.39013	0.695063	−	0.718949i	$-0.255377\pi$
$827$	39.9767	1.39013	0.695063	+	0.718949i	$0.255377\pi$
$828$	0	0
$829$	−29.1065	−1.01091	−0.505456	−	0.862852i	$-0.668676\pi$
$829$	−29.1065	−1.01091	−0.505456	+	0.862852i	$0.668676\pi$
$830$	0	0
$831$	−17.0526	−0.591549
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	14.0917	0.487664
$836$	0	0
$837$	−27.0755	−0.935868
$838$	0	0
$839$	53.7143	1.85442	0.927211	−	0.374539i	$-0.122199\pi$
$839$	53.7143	1.85442	0.927211	+	0.374539i	$0.122199\pi$
$840$	0	0
$841$	18.4854	0.637428
$842$	0	0
$843$	−7.65668	−0.263710
$844$	0	0
$845$	−9.08217	−0.312436
$846$	0	0
$847$	9.34739	0.321180
$848$	0	0
$849$	4.23259	0.145262
$850$	0	0
$851$	−12.1283	−0.415751
$852$	0	0
$853$	−3.46654	−0.118692	−0.0593461	−	0.998237i	$-0.518902\pi$
$853$	−3.46654	−0.118692	−0.0593461	+	0.998237i	$0.518902\pi$
$854$	0	0
$855$	−10.0470	−0.343602
$856$	0	0
$857$	49.3139	1.68453	0.842266	−	0.539062i	$-0.181221\pi$
$857$	49.3139	1.68453	0.842266	+	0.539062i	$0.181221\pi$
$858$	0	0
$859$	−29.0076	−0.989727	−0.494864	−	0.868971i	$-0.664782\pi$
$859$	−29.0076	−0.989727	−0.494864	+	0.868971i	$0.664782\pi$
$860$	0	0
$861$	1.25885	0.0429015
$862$	0	0
$863$	−36.0564	−1.22738	−0.613688	−	0.789549i	$-0.710315\pi$
$863$	−36.0564	−1.22738	−0.613688	+	0.789549i	$0.710315\pi$
$864$	0	0
$865$	4.69011	0.159468
$866$	0	0
$867$	−0.783617	−0.0266130
$868$	0	0
$869$	−11.3806	−0.386060
$870$	0	0
$871$	−12.0477	−0.408222
$872$	0	0
$873$	31.5464	1.06769
$874$	0	0
$875$	7.23238	0.244499
$876$	0	0
$877$	48.5717	1.64015	0.820075	−	0.572256i	$-0.193931\pi$
$877$	48.5717	1.64015	0.820075	+	0.572256i	$0.193931\pi$
$878$	0	0
$879$	2.40215	0.0810224
$880$	0	0
$881$	−53.1509	−1.79070	−0.895349	−	0.445365i	$-0.853074\pi$
$881$	−53.1509	−1.79070	−0.895349	+	0.445365i	$0.853074\pi$
$882$	0	0
$883$	4.45366	0.149878	0.0749388	−	0.997188i	$-0.476124\pi$
$883$	4.45366	0.149878	0.0749388	+	0.997188i	$0.476124\pi$
$884$	0	0
$885$	−5.00487	−0.168237
$886$	0	0
$887$	19.4244	0.652206	0.326103	−	0.945334i	$-0.394264\pi$
$887$	19.4244	0.652206	0.326103	+	0.945334i	$0.394264\pi$
$888$	0	0
$889$	−12.2550	−0.411020
$890$	0	0
$891$	−4.95006	−0.165833
$892$	0	0
$893$	28.4074	0.950619
$894$	0	0
$895$	3.40527	0.113826
$896$	0	0
$897$	−5.43540	−0.181483
$898$	0	0
$899$	44.2071	1.47439
$900$	0	0
$901$	9.60064	0.319844
$902$	0	0
$903$	4.17534	0.138947
$904$	0	0
$905$	9.73283	0.323530
$906$	0	0
$907$	52.6769	1.74911	0.874555	−	0.484927i	$-0.161154\pi$
$907$	52.6769	1.74911	0.874555	+	0.484927i	$0.161154\pi$
$908$	0	0
$909$	−17.3029	−0.573900
$910$	0	0
$911$	19.2458	0.637642	0.318821	−	0.947815i	$-0.396713\pi$
$911$	19.2458	0.637642	0.318821	+	0.947815i	$0.396713\pi$
$912$	0	0
$913$	6.84840	0.226649
$914$	0	0
$915$	−4.66490	−0.154217
$916$	0	0
$917$	7.87643	0.260103
$918$	0	0
$919$	−46.9278	−1.54801	−0.774003	−	0.633182i	$-0.781749\pi$
$919$	−46.9278	−1.54801	−0.774003	+	0.633182i	$0.781749\pi$
$920$	0	0
$921$	20.9821	0.691383
$922$	0	0
$923$	5.84386	0.192353
$924$	0	0
$925$	8.38992	0.275859
$926$	0	0
$927$	−14.5934	−0.479310
$928$	0	0
$929$	−15.0255	−0.492970	−0.246485	−	0.969147i	$-0.579276\pi$
$929$	−15.0255	−0.492970	−0.246485	+	0.969147i	$0.579276\pi$
$930$	0	0
$931$	5.47833	0.179545
$932$	0	0
$933$	−18.4984	−0.605611
$934$	0	0
$935$	0.988133	0.0323154
$936$	0	0
$937$	−30.9575	−1.01134	−0.505669	−	0.862728i	$-0.668754\pi$
$937$	−30.9575	−1.01134	−0.505669	+	0.862728i	$0.668754\pi$
$938$	0	0
$939$	−0.523873	−0.0170960
$940$	0	0
$941$	−41.8333	−1.36373	−0.681863	−	0.731480i	$-0.738830\pi$
$941$	−41.8333	−1.36373	−0.681863	+	0.731480i	$0.738830\pi$
$942$	0	0
$943$	10.2392	0.333436
$944$	0	0
$945$	−3.24411	−0.105531
$946$	0	0
$947$	−2.96865	−0.0964682	−0.0482341	−	0.998836i	$-0.515359\pi$
$947$	−2.96865	−0.0964682	−0.0482341	+	0.998836i	$0.515359\pi$
$948$	0	0
$949$	9.81337	0.318556
$950$	0	0
$951$	8.27386	0.268298
$952$	0	0
$953$	31.6462	1.02512	0.512560	−	0.858651i	$-0.328697\pi$
$953$	31.6462	1.02512	0.512560	+	0.858651i	$0.328697\pi$
$954$	0	0
$955$	−5.86458	−0.189773
$956$	0	0
$957$	−6.94175	−0.224395
$958$	0	0
$959$	−17.8546	−0.576555
$960$	0	0
$961$	10.1551	0.327584
$962$	0	0
$963$	−10.6866	−0.344371
$964$	0	0
$965$	−5.77370	−0.185862
$966$	0	0
$967$	−35.7671	−1.15019	−0.575097	−	0.818085i	$-0.695036\pi$
$967$	−35.7671	−1.15019	−0.575097	+	0.818085i	$0.695036\pi$
$968$	0	0
$969$	4.29291	0.137908
$970$	0	0
$971$	29.1248	0.934660	0.467330	−	0.884083i	$-0.345216\pi$
$971$	29.1248	0.934660	0.467330	+	0.884083i	$0.345216\pi$
$972$	0	0
$973$	−12.1290	−0.388839
$974$	0	0
$975$	3.76002	0.120417
$976$	0	0
$977$	52.2733	1.67237	0.836185	−	0.548448i	$-0.184781\pi$
$977$	52.2733	1.67237	0.836185	+	0.548448i	$0.184781\pi$
$978$	0	0
$979$	−5.60514	−0.179141
$980$	0	0
$981$	−44.8373	−1.43154
$982$	0	0
$983$	−23.0540	−0.735308	−0.367654	−	0.929963i	$-0.619839\pi$
$983$	−23.0540	−0.735308	−0.367654	+	0.929963i	$0.619839\pi$
$984$	0	0
$985$	−5.27602	−0.168108
$986$	0	0
$987$	4.06338	0.129339
$988$	0	0
$989$	33.9614	1.07991
$990$	0	0
$991$	−0.246696	−0.00783657	−0.00391829	−	0.999992i	$-0.501247\pi$
$991$	−0.246696	−0.00783657	−0.00391829	+	0.999992i	$0.501247\pi$
$992$	0	0
$993$	−9.42793	−0.299186
$994$	0	0
$995$	−4.90360	−0.155455
$996$	0	0
$997$	22.5050	0.712741	0.356371	−	0.934345i	$-0.384014\pi$
$997$	22.5050	0.712741	0.356371	+	0.934345i	$0.384014\pi$
$998$	0	0
$999$	−8.03093	−0.254088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.e.1.2	✓	5
4.3	odd	2		3808.2.a.f.1.4	yes	5
8.3	odd	2		7616.2.a.bs.1.2		5
8.5	even	2		7616.2.a.br.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.e.1.2	✓	5	1.1	even	1	trivial
3808.2.a.f.1.4	yes	5	4.3	odd	2
7616.2.a.br.1.4		5	8.5	even	2
7616.2.a.bs.1.2		5	8.3	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:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q-0.783617 q^{3} +0.768652 q^{5} -1.00000 q^{7} -2.38595 q^{9} -1.28554 q^{11} +1.08825 q^{13} -0.602329 q^{15} -1.00000 q^{17} +5.47833 q^{19} +0.783617 q^{21} +6.37379 q^{23} -4.40917 q^{25} +4.22052 q^{27} -6.89097 q^{29} -6.41522 q^{31} +1.00737 q^{33} -0.768652 q^{35} -1.90283 q^{37} -0.852773 q^{39} +1.60646 q^{41} +5.32829 q^{43} -1.83396 q^{45} +5.18542 q^{47} +1.00000 q^{49} +0.783617 q^{51} -9.60064 q^{53} -0.988133 q^{55} -4.29291 q^{57} +8.30919 q^{59} +7.74477 q^{61} +2.38595 q^{63} +0.836488 q^{65} -11.0707 q^{67} -4.99461 q^{69} +5.36995 q^{71} +9.01755 q^{73} +3.45510 q^{75} +1.28554 q^{77} +8.85277 q^{79} +3.85057 q^{81} -5.32726 q^{83} -0.768652 q^{85} +5.39987 q^{87} +4.36015 q^{89} -1.08825 q^{91} +5.02707 q^{93} +4.21093 q^{95} -13.2218 q^{97} +3.06723 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 5 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 8 q^{15} - 5 q^{17} - 6 q^{19} + 18 q^{23} + 5 q^{25} + 18 q^{27} - 14 q^{29} + 16 q^{31} - 26 q^{33} + 2 q^{35} + 2 q^{37} + 6 q^{39} - 10 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^5 - 5 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 8 * q^15 - 5 * q^17 - 6 * q^19 + 18 * q^23 + 5 * q^25 + 18 * q^27 - 14 * q^29 + 16 * q^31 - 26 * q^33 + 2 * q^35 + 2 * q^37 + 6 * q^39 - 10 * q^41 - 20 * q^43 + 6 * q^45 + 26 * q^47 + 5 * q^49 + 4 * q^55 + 12 * q^57 + 8 * q^61 - 3 * q^63 + 2 * q^65 + 12 * q^67 + 32 * q^69 + 10 * q^71 - 14 * q^73 - 4 * q^75 + 4 * q^77 + 34 * q^79 + 33 * q^81 + 20 * q^83 + 2 * q^85 + 14 * q^87 - 4 * q^89 + 6 * q^91 + 16 * q^93 + 42 * q^95 - 12 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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