LMFDB - Embedded newform 9680.2.a.cm.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,-3,0,0,0,0,0,1,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.09529$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.323071 q^{3} -1.00000 q^{5} +2.68522 q^{7} -2.89563 q^{9} +4.66785 q^{13} +0.323071 q^{15} +4.62632 q^{17} -4.34044 q^{19} -0.867517 q^{21} -2.77222 q^{23} +1.00000 q^{25} +1.90471 q^{27} +3.01341 q^{29} -2.38630 q^{31} -2.68522 q^{35} -10.6429 q^{37} -1.50805 q^{39} +2.21041 q^{41} -7.06719 q^{43} +2.89563 q^{45} -4.36215 q^{47} +0.210405 q^{49} -1.49463 q^{51} -6.33404 q^{53} +1.40227 q^{57} -11.7473 q^{59} +3.98263 q^{61} -7.77539 q^{63} -4.66785 q^{65} -7.31984 q^{67} +0.895625 q^{69} -1.19571 q^{71} -1.02171 q^{73} -0.323071 q^{75} +3.50213 q^{79} +8.07152 q^{81} +11.1158 q^{83} -4.62632 q^{85} -0.973547 q^{87} +2.76978 q^{89} +12.5342 q^{91} +0.770945 q^{93} +4.34044 q^{95} +18.5342 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 3 q^{7} + q^{13} - q^{17} - 20 q^{19} + 10 q^{21} - 5 q^{23} + 4 q^{25} + 15 q^{27} + 12 q^{29} + 5 q^{31} + 3 q^{35} + 7 q^{37} - 7 q^{39} + 11 q^{41} - 19 q^{43} - 5 q^{47} + 3 q^{49}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 3 * q^7 + q^13 - q^17 - 20 * q^19 + 10 * q^21 - 5 * q^23 + 4 * q^25 + 15 * q^27 + 12 * q^29 + 5 * q^31 + 3 * q^35 + 7 * q^37 - 7 * q^39 + 11 * q^41 - 19 * q^43 - 5 * q^47 + 3 * q^49 - 7 * q^51 - 11 * q^53 - 5 * q^57 - 9 * q^59 + 12 * q^61 + 5 * q^63 - q^65 + 19 * q^67 - 8 * q^69 - 5 * q^71 + 11 * q^73 - 34 * q^79 + 4 * q^81 + 11 * q^83 + q^85 + 19 * q^87 - 8 * q^89 + 8 * q^91 - 5 * q^93 + 20 * q^95 + 32 * q^97

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
$3$	−0.323071	−0.186525	−0.0932626	−	0.995642i	$-0.529730\pi$
$3$	−0.323071	−0.186525	−0.0932626	+	0.995642i	$0.529730\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.68522	1.01492	0.507459	−	0.861676i	$-0.330585\pi$
$7$	2.68522	1.01492	0.507459	+	0.861676i	$0.330585\pi$
$8$	0	0
$9$	−2.89563	−0.965208
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.66785	1.29463	0.647314	−	0.762223i	$-0.275892\pi$
$13$	4.66785	1.29463	0.647314	+	0.762223i	$0.275892\pi$
$14$	0	0
$15$	0.323071	0.0834166
$16$	0	0
$17$	4.62632	1.12205	0.561024	−	0.827799i	$-0.310407\pi$
$17$	4.62632	1.12205	0.561024	+	0.827799i	$0.310407\pi$
$18$	0	0
$19$	−4.34044	−0.995766	−0.497883	−	0.867244i	$-0.665889\pi$
$19$	−4.34044	−0.995766	−0.497883	+	0.867244i	$0.665889\pi$
$20$	0	0
$21$	−0.867517	−0.189308
$22$	0	0
$23$	−2.77222	−0.578048	−0.289024	−	0.957322i	$-0.593331\pi$
$23$	−2.77222	−0.578048	−0.289024	+	0.957322i	$0.593331\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.90471	0.366561
$28$	0	0
$29$	3.01341	0.559577	0.279789	−	0.960062i	$-0.409736\pi$
$29$	3.01341	0.559577	0.279789	+	0.960062i	$0.409736\pi$
$30$	0	0
$31$	−2.38630	−0.428592	−0.214296	−	0.976769i	$-0.568746\pi$
$31$	−2.38630	−0.428592	−0.214296	+	0.976769i	$0.568746\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.68522	−0.453885
$36$	0	0
$37$	−10.6429	−1.74968	−0.874842	−	0.484409i	$-0.839035\pi$
$37$	−10.6429	−1.74968	−0.874842	+	0.484409i	$0.839035\pi$
$38$	0	0
$39$	−1.50805	−0.241481
$40$	0	0
$41$	2.21041	0.345207	0.172604	−	0.984991i	$-0.444782\pi$
$41$	2.21041	0.345207	0.172604	+	0.984991i	$0.444782\pi$
$42$	0	0
$43$	−7.06719	−1.07774	−0.538868	−	0.842390i	$-0.681148\pi$
$43$	−7.06719	−1.07774	−0.538868	+	0.842390i	$0.681148\pi$
$44$	0	0
$45$	2.89563	0.431654
$46$	0	0
$47$	−4.36215	−0.636285	−0.318142	−	0.948043i	$-0.603059\pi$
$47$	−4.36215	−0.636285	−0.318142	+	0.948043i	$0.603059\pi$
$48$	0	0
$49$	0.210405	0.0300579
$50$	0	0
$51$	−1.49463	−0.209290
$52$	0	0
$53$	−6.33404	−0.870047	−0.435024	−	0.900419i	$-0.643260\pi$
$53$	−6.33404	−0.870047	−0.435024	+	0.900419i	$0.643260\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.40227	0.185735
$58$	0	0
$59$	−11.7473	−1.52937	−0.764683	−	0.644407i	$-0.777104\pi$
$59$	−11.7473	−1.52937	−0.764683	+	0.644407i	$0.777104\pi$
$60$	0	0
$61$	3.98263	0.509923	0.254962	−	0.966951i	$-0.417937\pi$
$61$	3.98263	0.509923	0.254962	+	0.966951i	$0.417937\pi$
$62$	0	0
$63$	−7.77539	−0.979607
$64$	0	0
$65$	−4.66785	−0.578975
$66$	0	0
$67$	−7.31984	−0.894260	−0.447130	−	0.894469i	$-0.647554\pi$
$67$	−7.31984	−0.894260	−0.447130	+	0.894469i	$0.647554\pi$
$68$	0	0
$69$	0.895625	0.107821
$70$	0	0
$71$	−1.19571	−0.141905	−0.0709525	−	0.997480i	$-0.522604\pi$
$71$	−1.19571	−0.141905	−0.0709525	+	0.997480i	$0.522604\pi$
$72$	0	0
$73$	−1.02171	−0.119582	−0.0597908	−	0.998211i	$-0.519043\pi$
$73$	−1.02171	−0.119582	−0.0597908	+	0.998211i	$0.519043\pi$
$74$	0	0
$75$	−0.323071	−0.0373050
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.50213	0.394021	0.197010	−	0.980401i	$-0.436877\pi$
$79$	3.50213	0.394021	0.197010	+	0.980401i	$0.436877\pi$
$80$	0	0
$81$	8.07152	0.896836
$82$	0	0
$83$	11.1158	1.22012	0.610061	−	0.792355i	$-0.291145\pi$
$83$	11.1158	1.22012	0.610061	+	0.792355i	$0.291145\pi$
$84$	0	0
$85$	−4.62632	−0.501795
$86$	0	0
$87$	−0.973547	−0.104375
$88$	0	0
$89$	2.76978	0.293596	0.146798	−	0.989167i	$-0.453103\pi$
$89$	2.76978	0.293596	0.146798	+	0.989167i	$0.453103\pi$
$90$	0	0
$91$	12.5342	1.31394
$92$	0	0
$93$	0.770945	0.0799432
$94$	0	0
$95$	4.34044	0.445320
$96$	0	0
$97$	18.5342	1.88186	0.940931	−	0.338597i	$-0.109952\pi$
$97$	18.5342	1.88186	0.940931	+	0.338597i	$0.109952\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.11455	0.707925	0.353962	−	0.935260i	$-0.384834\pi$
$101$	7.11455	0.707925	0.353962	+	0.935260i	$0.384834\pi$
$102$	0	0
$103$	−7.52835	−0.741791	−0.370895	−	0.928675i	$-0.620949\pi$
$103$	−7.52835	−0.741791	−0.370895	+	0.928675i	$0.620949\pi$
$104$	0	0
$105$	0.867517	0.0846610
$106$	0	0
$107$	18.0292	1.74295	0.871475	−	0.490441i	$-0.163164\pi$
$107$	18.0292	1.74295	0.871475	+	0.490441i	$0.163164\pi$
$108$	0	0
$109$	16.3653	1.56751	0.783756	−	0.621068i	$-0.213301\pi$
$109$	16.3653	1.56751	0.783756	+	0.621068i	$0.213301\pi$
$110$	0	0
$111$	3.43842	0.326360
$112$	0	0
$113$	−2.05377	−0.193203	−0.0966013	−	0.995323i	$-0.530797\pi$
$113$	−2.05377	−0.193203	−0.0966013	+	0.995323i	$0.530797\pi$
$114$	0	0
$115$	2.77222	0.258511
$116$	0	0
$117$	−13.5163	−1.24959
$118$	0	0
$119$	12.4227	1.13879
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.714118	−0.0643899
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.0762667	−0.00676758	−0.00338379	−	0.999994i	$-0.501077\pi$
$127$	−0.0762667	−0.00676758	−0.00338379	+	0.999994i	$0.501077\pi$
$128$	0	0
$129$	2.28320	0.201025
$130$	0	0
$131$	−11.4831	−1.00328	−0.501642	−	0.865075i	$-0.667270\pi$
$131$	−11.4831	−1.00328	−0.501642	+	0.865075i	$0.667270\pi$
$132$	0	0
$133$	−11.6550	−1.01062
$134$	0	0
$135$	−1.90471	−0.163931
$136$	0	0
$137$	−18.3293	−1.56598	−0.782989	−	0.622036i	$-0.786306\pi$
$137$	−18.3293	−1.56598	−0.782989	+	0.622036i	$0.786306\pi$
$138$	0	0
$139$	−23.1874	−1.96673	−0.983363	−	0.181653i	$-0.941855\pi$
$139$	−23.1874	−1.96673	−0.983363	+	0.181653i	$0.941855\pi$
$140$	0	0
$141$	1.40928	0.118683
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.01341	−0.250250
$146$	0	0
$147$	−0.0679759	−0.00560655
$148$	0	0
$149$	−14.6646	−1.20137	−0.600686	−	0.799485i	$-0.705106\pi$
$149$	−14.6646	−1.20137	−0.600686	+	0.799485i	$0.705106\pi$
$150$	0	0
$151$	−7.51902	−0.611889	−0.305944	−	0.952049i	$-0.598972\pi$
$151$	−7.51902	−0.611889	−0.305944	+	0.952049i	$0.598972\pi$
$152$	0	0
$153$	−13.3961	−1.08301
$154$	0	0
$155$	2.38630	0.191672
$156$	0	0
$157$	−13.3819	−1.06799	−0.533996	−	0.845487i	$-0.679310\pi$
$157$	−13.3819	−1.06799	−0.533996	+	0.845487i	$0.679310\pi$
$158$	0	0
$159$	2.04635	0.162286
$160$	0	0
$161$	−7.44403	−0.586672
$162$	0	0
$163$	0.771990	0.0604669	0.0302335	−	0.999543i	$-0.490375\pi$
$163$	0.771990	0.0604669	0.0302335	+	0.999543i	$0.490375\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.48232	−0.656381	−0.328191	−	0.944612i	$-0.606439\pi$
$167$	−8.48232	−0.656381	−0.328191	+	0.944612i	$0.606439\pi$
$168$	0	0
$169$	8.78880	0.676062
$170$	0	0
$171$	12.5683	0.961122
$172$	0	0
$173$	−5.07871	−0.386127	−0.193064	−	0.981186i	$-0.561842\pi$
$173$	−5.07871	−0.386127	−0.193064	+	0.981186i	$0.561842\pi$
$174$	0	0
$175$	2.68522	0.202984
$176$	0	0
$177$	3.79521	0.285265
$178$	0	0
$179$	11.3170	0.845876	0.422938	−	0.906159i	$-0.360999\pi$
$179$	11.3170	0.845876	0.422938	+	0.906159i	$0.360999\pi$
$180$	0	0
$181$	7.40006	0.550042	0.275021	−	0.961438i	$-0.411315\pi$
$181$	7.40006	0.550042	0.275021	+	0.961438i	$0.411315\pi$
$182$	0	0
$183$	−1.28667	−0.0951135
$184$	0	0
$185$	10.6429	0.782482
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.11455	0.372029
$190$	0	0
$191$	5.15608	0.373081	0.186540	−	0.982447i	$-0.440272\pi$
$191$	5.15608	0.373081	0.186540	+	0.982447i	$0.440272\pi$
$192$	0	0
$193$	4.03230	0.290251	0.145126	−	0.989413i	$-0.453641\pi$
$193$	4.03230	0.290251	0.145126	+	0.989413i	$0.453641\pi$
$194$	0	0
$195$	1.50805	0.107993
$196$	0	0
$197$	−11.4176	−0.813469	−0.406734	−	0.913547i	$-0.633333\pi$
$197$	−11.4176	−0.813469	−0.406734	+	0.913547i	$0.633333\pi$
$198$	0	0
$199$	7.16644	0.508015	0.254008	−	0.967202i	$-0.418251\pi$
$199$	7.16644	0.508015	0.254008	+	0.967202i	$0.418251\pi$
$200$	0	0
$201$	2.36483	0.166802
$202$	0	0
$203$	8.09168	0.567925
$204$	0	0
$205$	−2.21041	−0.154381
$206$	0	0
$207$	8.02732	0.557937
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.48359	0.239820	0.119910	−	0.992785i	$-0.461739\pi$
$211$	3.48359	0.239820	0.119910	+	0.992785i	$0.461739\pi$
$212$	0	0
$213$	0.386300	0.0264688
$214$	0	0
$215$	7.06719	0.481978
$216$	0	0
$217$	−6.40774	−0.434986
$218$	0	0
$219$	0.330084	0.0223050
$220$	0	0
$221$	21.5950	1.45264
$222$	0	0
$223$	−10.5449	−0.706141	−0.353071	−	0.935597i	$-0.614862\pi$
$223$	−10.5449	−0.706141	−0.353071	+	0.935597i	$0.614862\pi$
$224$	0	0
$225$	−2.89563	−0.193042
$226$	0	0
$227$	0.216018	0.0143376	0.00716880	−	0.999974i	$-0.497718\pi$
$227$	0.216018	0.0143376	0.00716880	+	0.999974i	$0.497718\pi$
$228$	0	0
$229$	−0.0757097	−0.00500304	−0.00250152	−	0.999997i	$-0.500796\pi$
$229$	−0.0757097	−0.00500304	−0.00250152	+	0.999997i	$0.500796\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.1201	−0.990548	−0.495274	−	0.868737i	$-0.664932\pi$
$233$	−15.1201	−0.990548	−0.495274	+	0.868737i	$0.664932\pi$
$234$	0	0
$235$	4.36215	0.284555
$236$	0	0
$237$	−1.13144	−0.0734948
$238$	0	0
$239$	−23.1646	−1.49839	−0.749197	−	0.662347i	$-0.769561\pi$
$239$	−23.1646	−1.49839	−0.749197	+	0.662347i	$0.769561\pi$
$240$	0	0
$241$	21.3349	1.37430	0.687151	−	0.726515i	$-0.258861\pi$
$241$	21.3349	1.37430	0.687151	+	0.726515i	$0.258861\pi$
$242$	0	0
$243$	−8.32179	−0.533843
$244$	0	0
$245$	−0.210405	−0.0134423
$246$	0	0
$247$	−20.2605	−1.28915
$248$	0	0
$249$	−3.59120	−0.227583
$250$	0	0
$251$	6.22186	0.392721	0.196360	−	0.980532i	$-0.437088\pi$
$251$	6.22186	0.392721	0.196360	+	0.980532i	$0.437088\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.49463	0.0935975
$256$	0	0
$257$	−14.2903	−0.891406	−0.445703	−	0.895181i	$-0.647046\pi$
$257$	−14.2903	−0.891406	−0.445703	+	0.895181i	$0.647046\pi$
$258$	0	0
$259$	−28.5785	−1.77578
$260$	0	0
$261$	−8.72572	−0.540108
$262$	0	0
$263$	4.13132	0.254748	0.127374	−	0.991855i	$-0.459345\pi$
$263$	4.13132	0.254748	0.127374	+	0.991855i	$0.459345\pi$
$264$	0	0
$265$	6.33404	0.389097
$266$	0	0
$267$	−0.894835	−0.0547630
$268$	0	0
$269$	−1.68394	−0.102672	−0.0513359	−	0.998681i	$-0.516348\pi$
$269$	−1.68394	−0.102672	−0.0513359	+	0.998681i	$0.516348\pi$
$270$	0	0
$271$	−18.4310	−1.11960	−0.559801	−	0.828627i	$-0.689123\pi$
$271$	−18.4310	−1.11960	−0.559801	+	0.828627i	$0.689123\pi$
$272$	0	0
$273$	−4.04944	−0.245083
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.42745	0.205935	0.102968	−	0.994685i	$-0.467166\pi$
$277$	3.42745	0.205935	0.102968	+	0.994685i	$0.467166\pi$
$278$	0	0
$279$	6.90983	0.413681
$280$	0	0
$281$	22.8217	1.36143	0.680715	−	0.732548i	$-0.261669\pi$
$281$	22.8217	1.36143	0.680715	+	0.732548i	$0.261669\pi$
$282$	0	0
$283$	−29.0991	−1.72976	−0.864880	−	0.501978i	$-0.832606\pi$
$283$	−29.0991	−1.72976	−0.864880	+	0.501978i	$0.832606\pi$
$284$	0	0
$285$	−1.40227	−0.0830634
$286$	0	0
$287$	5.93542	0.350357
$288$	0	0
$289$	4.40288	0.258993
$290$	0	0
$291$	−5.98786	−0.351015
$292$	0	0
$293$	−21.2136	−1.23931	−0.619655	−	0.784874i	$-0.712727\pi$
$293$	−21.2136	−1.23931	−0.619655	+	0.784874i	$0.712727\pi$
$294$	0	0
$295$	11.7473	0.683953
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.9403	−0.748358
$300$	0	0
$301$	−18.9769	−1.09381
$302$	0	0
$303$	−2.29851	−0.132046
$304$	0	0
$305$	−3.98263	−0.228045
$306$	0	0
$307$	−6.87520	−0.392388	−0.196194	−	0.980565i	$-0.562858\pi$
$307$	−6.87520	−0.392388	−0.196194	+	0.980565i	$0.562858\pi$
$308$	0	0
$309$	2.43219	0.138363
$310$	0	0
$311$	−25.1577	−1.42656	−0.713280	−	0.700879i	$-0.752791\pi$
$311$	−25.1577	−1.42656	−0.713280	+	0.700879i	$0.752791\pi$
$312$	0	0
$313$	−11.5793	−0.654503	−0.327251	−	0.944937i	$-0.606122\pi$
$313$	−11.5793	−0.654503	−0.327251	+	0.944937i	$0.606122\pi$
$314$	0	0
$315$	7.77539	0.438094
$316$	0	0
$317$	20.7413	1.16495	0.582473	−	0.812850i	$-0.302085\pi$
$317$	20.7413	1.16495	0.582473	+	0.812850i	$0.302085\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.82472	−0.325104
$322$	0	0
$323$	−20.0803	−1.11730
$324$	0	0
$325$	4.66785	0.258926
$326$	0	0
$327$	−5.28716	−0.292381
$328$	0	0
$329$	−11.7133	−0.645777
$330$	0	0
$331$	32.1415	1.76665	0.883327	−	0.468757i	$-0.155298\pi$
$331$	32.1415	1.76665	0.883327	+	0.468757i	$0.155298\pi$
$332$	0	0
$333$	30.8179	1.68881
$334$	0	0
$335$	7.31984	0.399925
$336$	0	0
$337$	17.9964	0.980326	0.490163	−	0.871631i	$-0.336937\pi$
$337$	17.9964	0.980326	0.490163	+	0.871631i	$0.336937\pi$
$338$	0	0
$339$	0.663514	0.0360371
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.2316	−0.984411
$344$	0	0
$345$	−0.895625	−0.0482188
$346$	0	0
$347$	−8.04981	−0.432137	−0.216068	−	0.976378i	$-0.569323\pi$
$347$	−8.04981	−0.432137	−0.216068	+	0.976378i	$0.569323\pi$
$348$	0	0
$349$	−19.2294	−1.02932	−0.514662	−	0.857393i	$-0.672083\pi$
$349$	−19.2294	−1.02932	−0.514662	+	0.857393i	$0.672083\pi$
$350$	0	0
$351$	8.89088	0.474560
$352$	0	0
$353$	14.8497	0.790371	0.395186	−	0.918601i	$-0.370680\pi$
$353$	14.8497	0.790371	0.395186	+	0.918601i	$0.370680\pi$
$354$	0	0
$355$	1.19571	0.0634618
$356$	0	0
$357$	−4.01341	−0.212412
$358$	0	0
$359$	10.2233	0.539563	0.269782	−	0.962922i	$-0.413048\pi$
$359$	10.2233	0.539563	0.269782	+	0.962922i	$0.413048\pi$
$360$	0	0
$361$	−0.160555	−0.00845028
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.02171	0.0534785
$366$	0	0
$367$	−14.1434	−0.738279	−0.369139	−	0.929374i	$-0.620347\pi$
$367$	−14.1434	−0.738279	−0.369139	+	0.929374i	$0.620347\pi$
$368$	0	0
$369$	−6.40050	−0.333197
$370$	0	0
$371$	−17.0083	−0.883026
$372$	0	0
$373$	−12.4600	−0.645154	−0.322577	−	0.946543i	$-0.604549\pi$
$373$	−12.4600	−0.645154	−0.322577	+	0.946543i	$0.604549\pi$
$374$	0	0
$375$	0.323071	0.0166833
$376$	0	0
$377$	14.0662	0.724444
$378$	0	0
$379$	−16.3370	−0.839174	−0.419587	−	0.907715i	$-0.637825\pi$
$379$	−16.3370	−0.839174	−0.419587	+	0.907715i	$0.637825\pi$
$380$	0	0
$381$	0.0246396	0.00126232
$382$	0	0
$383$	0.812648	0.0415244	0.0207622	−	0.999784i	$-0.493391\pi$
$383$	0.812648	0.0415244	0.0207622	+	0.999784i	$0.493391\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.4639	1.04024
$388$	0	0
$389$	−30.3941	−1.54104	−0.770521	−	0.637414i	$-0.780004\pi$
$389$	−30.3941	−1.54104	−0.770521	+	0.637414i	$0.780004\pi$
$390$	0	0
$391$	−12.8252	−0.648598
$392$	0	0
$393$	3.70986	0.187138
$394$	0	0
$395$	−3.50213	−0.176211
$396$	0	0
$397$	−14.8996	−0.747789	−0.373894	−	0.927471i	$-0.621978\pi$
$397$	−14.8996	−0.747789	−0.373894	+	0.927471i	$0.621978\pi$
$398$	0	0
$399$	3.76541	0.188506
$400$	0	0
$401$	12.1692	0.607700	0.303850	−	0.952720i	$-0.401728\pi$
$401$	12.1692	0.607700	0.303850	+	0.952720i	$0.401728\pi$
$402$	0	0
$403$	−11.1389	−0.554867
$404$	0	0
$405$	−8.07152	−0.401077
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.262108	0.0129604	0.00648020	−	0.999979i	$-0.497937\pi$
$409$	0.262108	0.0129604	0.00648020	+	0.999979i	$0.497937\pi$
$410$	0	0
$411$	5.92166	0.292094
$412$	0	0
$413$	−31.5440	−1.55218
$414$	0	0
$415$	−11.1158	−0.545655
$416$	0	0
$417$	7.49116	0.366844
$418$	0	0
$419$	1.26916	0.0620023	0.0310012	−	0.999519i	$-0.490130\pi$
$419$	1.26916	0.0620023	0.0310012	+	0.999519i	$0.490130\pi$
$420$	0	0
$421$	−29.6655	−1.44581	−0.722903	−	0.690949i	$-0.757193\pi$
$421$	−29.6655	−1.44581	−0.722903	+	0.690949i	$0.757193\pi$
$422$	0	0
$423$	12.6311	0.614147
$424$	0	0
$425$	4.62632	0.224410
$426$	0	0
$427$	10.6942	0.517530
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.3549	1.51031	0.755156	−	0.655545i	$-0.227561\pi$
$431$	31.3549	1.51031	0.755156	+	0.655545i	$0.227561\pi$
$432$	0	0
$433$	−26.0325	−1.25104	−0.625521	−	0.780207i	$-0.715114\pi$
$433$	−26.0325	−1.25104	−0.625521	+	0.780207i	$0.715114\pi$
$434$	0	0
$435$	0.973547	0.0466780
$436$	0	0
$437$	12.0327	0.575601
$438$	0	0
$439$	14.4191	0.688185	0.344093	−	0.938936i	$-0.388187\pi$
$439$	14.4191	0.688185	0.344093	+	0.938936i	$0.388187\pi$
$440$	0	0
$441$	−0.609255	−0.0290121
$442$	0	0
$443$	0.330608	0.0157077	0.00785383	−	0.999969i	$-0.497500\pi$
$443$	0.330608	0.0157077	0.00785383	+	0.999969i	$0.497500\pi$
$444$	0	0
$445$	−2.76978	−0.131300
$446$	0	0
$447$	4.73771	0.224086
$448$	0	0
$449$	8.50828	0.401531	0.200765	−	0.979639i	$-0.435657\pi$
$449$	8.50828	0.401531	0.200765	+	0.979639i	$0.435657\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.42918	0.114133
$454$	0	0
$455$	−12.5342	−0.587612
$456$	0	0
$457$	−1.13847	−0.0532555	−0.0266277	−	0.999645i	$-0.508477\pi$
$457$	−1.13847	−0.0532555	−0.0266277	+	0.999645i	$0.508477\pi$
$458$	0	0
$459$	8.81179	0.411299
$460$	0	0
$461$	−14.5073	−0.675670	−0.337835	−	0.941205i	$-0.609695\pi$
$461$	−14.5073	−0.675670	−0.337835	+	0.941205i	$0.609695\pi$
$462$	0	0
$463$	4.89739	0.227601	0.113801	−	0.993504i	$-0.463698\pi$
$463$	4.89739	0.227601	0.113801	+	0.993504i	$0.463698\pi$
$464$	0	0
$465$	−0.770945	−0.0357517
$466$	0	0
$467$	−32.4230	−1.50036	−0.750180	−	0.661234i	$-0.770033\pi$
$467$	−32.4230	−1.50036	−0.750180	+	0.661234i	$0.770033\pi$
$468$	0	0
$469$	−19.6554	−0.907601
$470$	0	0
$471$	4.32330	0.199207
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.34044	−0.199153
$476$	0	0
$477$	18.3410	0.839777
$478$	0	0
$479$	17.7354	0.810352	0.405176	−	0.914239i	$-0.367210\pi$
$479$	17.7354	0.810352	0.405176	+	0.914239i	$0.367210\pi$
$480$	0	0
$481$	−49.6795	−2.26519
$482$	0	0
$483$	2.40495	0.109429
$484$	0	0
$485$	−18.5342	−0.841595
$486$	0	0
$487$	18.4603	0.836514	0.418257	−	0.908329i	$-0.362641\pi$
$487$	18.4603	0.836514	0.418257	+	0.908329i	$0.362641\pi$
$488$	0	0
$489$	−0.249408	−0.0112786
$490$	0	0
$491$	−11.4338	−0.516002	−0.258001	−	0.966145i	$-0.583064\pi$
$491$	−11.4338	−0.516002	−0.258001	+	0.966145i	$0.583064\pi$
$492$	0	0
$493$	13.9410	0.627873
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.21075	−0.144022
$498$	0	0
$499$	−10.8815	−0.487125	−0.243562	−	0.969885i	$-0.578316\pi$
$499$	−10.8815	−0.487125	−0.243562	+	0.969885i	$0.578316\pi$
$500$	0	0
$501$	2.74039	0.122432
$502$	0	0
$503$	44.7247	1.99417	0.997087	−	0.0762683i	$-0.0243005\pi$
$503$	44.7247	1.99417	0.997087	+	0.0762683i	$0.0243005\pi$
$504$	0	0
$505$	−7.11455	−0.316594
$506$	0	0
$507$	−2.83941	−0.126103
$508$	0	0
$509$	−13.8093	−0.612088	−0.306044	−	0.952017i	$-0.599006\pi$
$509$	−13.8093	−0.612088	−0.306044	+	0.952017i	$0.599006\pi$
$510$	0	0
$511$	−2.74350	−0.121365
$512$	0	0
$513$	−8.26727	−0.365009
$514$	0	0
$515$	7.52835	0.331739
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.64079	0.0720225
$520$	0	0
$521$	−3.64972	−0.159897	−0.0799486	−	0.996799i	$-0.525476\pi$
$521$	−3.64972	−0.159897	−0.0799486	+	0.996799i	$0.525476\pi$
$522$	0	0
$523$	−4.98707	−0.218069	−0.109035	−	0.994038i	$-0.534776\pi$
$523$	−4.98707	−0.218069	−0.109035	+	0.994038i	$0.534776\pi$
$524$	0	0
$525$	−0.867517	−0.0378615
$526$	0	0
$527$	−11.0398	−0.480901
$528$	0	0
$529$	−15.3148	−0.665860
$530$	0	0
$531$	34.0157	1.47616
$532$	0	0
$533$	10.3178	0.446915
$534$	0	0
$535$	−18.0292	−0.779471
$536$	0	0
$537$	−3.65621	−0.157777
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.247594	−0.0106449	−0.00532246	−	0.999986i	$-0.501694\pi$
$541$	−0.247594	−0.0106449	−0.00532246	+	0.999986i	$0.501694\pi$
$542$	0	0
$543$	−2.39075	−0.102597
$544$	0	0
$545$	−16.3653	−0.701013
$546$	0	0
$547$	−25.3120	−1.08226	−0.541132	−	0.840938i	$-0.682004\pi$
$547$	−25.3120	−1.08226	−0.541132	+	0.840938i	$0.682004\pi$
$548$	0	0
$549$	−11.5322	−0.492182
$550$	0	0
$551$	−13.0796	−0.557208
$552$	0	0
$553$	9.40400	0.399899
$554$	0	0
$555$	−3.43842	−0.145953
$556$	0	0
$557$	38.6671	1.63838	0.819190	−	0.573523i	$-0.194424\pi$
$557$	38.6671	1.63838	0.819190	+	0.573523i	$0.194424\pi$
$558$	0	0
$559$	−32.9885	−1.39527
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.9362	0.587341	0.293671	−	0.955907i	$-0.405123\pi$
$563$	13.9362	0.587341	0.293671	+	0.955907i	$0.405123\pi$
$564$	0	0
$565$	2.05377	0.0864028
$566$	0	0
$567$	21.6738	0.910214
$568$	0	0
$569$	−27.9125	−1.17015	−0.585076	−	0.810979i	$-0.698935\pi$
$569$	−27.9125	−1.17015	−0.585076	+	0.810979i	$0.698935\pi$
$570$	0	0
$571$	−31.4113	−1.31452	−0.657261	−	0.753663i	$-0.728285\pi$
$571$	−31.4113	−1.31452	−0.657261	+	0.753663i	$0.728285\pi$
$572$	0	0
$573$	−1.66578	−0.0695889
$574$	0	0
$575$	−2.77222	−0.115610
$576$	0	0
$577$	20.7349	0.863206	0.431603	−	0.902064i	$-0.357948\pi$
$577$	20.7349	0.863206	0.431603	+	0.902064i	$0.357948\pi$
$578$	0	0
$579$	−1.30272	−0.0541392
$580$	0	0
$581$	29.8485	1.23832
$582$	0	0
$583$	0	0
$584$	0	0
$585$	13.5163	0.558832
$586$	0	0
$587$	−15.2367	−0.628886	−0.314443	−	0.949276i	$-0.601818\pi$
$587$	−15.2367	−0.628886	−0.314443	+	0.949276i	$0.601818\pi$
$588$	0	0
$589$	10.3576	0.426777
$590$	0	0
$591$	3.68869	0.151732
$592$	0	0
$593$	−27.5413	−1.13098	−0.565492	−	0.824754i	$-0.691314\pi$
$593$	−27.5413	−1.13098	−0.565492	+	0.824754i	$0.691314\pi$
$594$	0	0
$595$	−12.4227	−0.509281
$596$	0	0
$597$	−2.31527	−0.0947576
$598$	0	0
$599$	26.7331	1.09228	0.546142	−	0.837693i	$-0.316096\pi$
$599$	26.7331	1.09228	0.546142	+	0.837693i	$0.316096\pi$
$600$	0	0
$601$	−2.40680	−0.0981753	−0.0490876	−	0.998794i	$-0.515631\pi$
$601$	−2.40680	−0.0981753	−0.0490876	+	0.998794i	$0.515631\pi$
$602$	0	0
$603$	21.1955	0.863147
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.2929	0.417774	0.208887	−	0.977940i	$-0.433016\pi$
$607$	10.2929	0.417774	0.208887	+	0.977940i	$0.433016\pi$
$608$	0	0
$609$	−2.61419	−0.105932
$610$	0	0
$611$	−20.3618	−0.823752
$612$	0	0
$613$	−27.8756	−1.12589	−0.562943	−	0.826496i	$-0.690331\pi$
$613$	−27.8756	−1.12589	−0.562943	+	0.826496i	$0.690331\pi$
$614$	0	0
$615$	0.714118	0.0287960
$616$	0	0
$617$	−28.7216	−1.15629	−0.578143	−	0.815935i	$-0.696222\pi$
$617$	−28.7216	−1.15629	−0.578143	+	0.815935i	$0.696222\pi$
$618$	0	0
$619$	−22.6968	−0.912261	−0.456131	−	0.889913i	$-0.650765\pi$
$619$	−22.6968	−0.912261	−0.456131	+	0.889913i	$0.650765\pi$
$620$	0	0
$621$	−5.28027	−0.211890
$622$	0	0
$623$	7.43746	0.297976
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−49.2375	−1.96323
$630$	0	0
$631$	−28.5364	−1.13602	−0.568008	−	0.823023i	$-0.692286\pi$
$631$	−28.5364	−1.13602	−0.568008	+	0.823023i	$0.692286\pi$
$632$	0	0
$633$	−1.12545	−0.0447326
$634$	0	0
$635$	0.0762667	0.00302655
$636$	0	0
$637$	0.982140	0.0389138
$638$	0	0
$639$	3.46233	0.136968
$640$	0	0
$641$	2.37375	0.0937575	0.0468788	−	0.998901i	$-0.485073\pi$
$641$	2.37375	0.0937575	0.0468788	+	0.998901i	$0.485073\pi$
$642$	0	0
$643$	−11.3603	−0.448007	−0.224004	−	0.974588i	$-0.571913\pi$
$643$	−11.3603	−0.448007	−0.224004	+	0.974588i	$0.571913\pi$
$644$	0	0
$645$	−2.28320	−0.0899010
$646$	0	0
$647$	48.4220	1.90366	0.951832	−	0.306621i	$-0.0991984\pi$
$647$	48.4220	1.90366	0.951832	+	0.306621i	$0.0991984\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.07016	0.0811358
$652$	0	0
$653$	29.7548	1.16440	0.582198	−	0.813047i	$-0.302193\pi$
$653$	29.7548	1.16440	0.582198	+	0.813047i	$0.302193\pi$
$654$	0	0
$655$	11.4831	0.448682
$656$	0	0
$657$	2.95848	0.115421
$658$	0	0
$659$	−28.4931	−1.10993	−0.554966	−	0.831873i	$-0.687269\pi$
$659$	−28.4931	−1.10993	−0.554966	+	0.831873i	$0.687269\pi$
$660$	0	0
$661$	−1.02875	−0.0400139	−0.0200070	−	0.999800i	$-0.506369\pi$
$661$	−1.02875	−0.0400139	−0.0200070	+	0.999800i	$0.506369\pi$
$662$	0	0
$663$	−6.97671	−0.270953
$664$	0	0
$665$	11.6550	0.451963
$666$	0	0
$667$	−8.35386	−0.323463
$668$	0	0
$669$	3.40676	0.131713
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−13.0369	−0.502536	−0.251268	−	0.967918i	$-0.580848\pi$
$673$	−13.0369	−0.502536	−0.251268	+	0.967918i	$0.580848\pi$
$674$	0	0
$675$	1.90471	0.0733122
$676$	0	0
$677$	37.1064	1.42612	0.713058	−	0.701105i	$-0.247310\pi$
$677$	37.1064	1.42612	0.713058	+	0.701105i	$0.247310\pi$
$678$	0	0
$679$	49.7684	1.90994
$680$	0	0
$681$	−0.0697890	−0.00267432
$682$	0	0
$683$	32.8992	1.25885	0.629426	−	0.777061i	$-0.283290\pi$
$683$	32.8992	1.25885	0.629426	+	0.777061i	$0.283290\pi$
$684$	0	0
$685$	18.3293	0.700326
$686$	0	0
$687$	0.0244596	0.000933193 0
$688$	0	0
$689$	−29.5663	−1.12639
$690$	0	0
$691$	−36.6998	−1.39613	−0.698064	−	0.716035i	$-0.745955\pi$
$691$	−36.6998	−1.39613	−0.698064	+	0.716035i	$0.745955\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.1874	0.879546
$696$	0	0
$697$	10.2261	0.387339
$698$	0	0
$699$	4.88485	0.184762
$700$	0	0
$701$	36.2497	1.36913	0.684566	−	0.728951i	$-0.259992\pi$
$701$	36.2497	1.36913	0.684566	+	0.728951i	$0.259992\pi$
$702$	0	0
$703$	46.1949	1.74227
$704$	0	0
$705$	−1.40928	−0.0530767
$706$	0	0
$707$	19.1041	0.718485
$708$	0	0
$709$	−18.6013	−0.698585	−0.349293	−	0.937014i	$-0.613578\pi$
$709$	−18.6013	−0.698585	−0.349293	+	0.937014i	$0.613578\pi$
$710$	0	0
$711$	−10.1409	−0.380312
$712$	0	0
$713$	6.61536	0.247747
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.48382	0.279488
$718$	0	0
$719$	37.4280	1.39583	0.697914	−	0.716181i	$-0.254112\pi$
$719$	37.4280	1.39583	0.697914	+	0.716181i	$0.254112\pi$
$720$	0	0
$721$	−20.2153	−0.752856
$722$	0	0
$723$	−6.89269	−0.256342
$724$	0	0
$725$	3.01341	0.111915
$726$	0	0
$727$	−14.6011	−0.541526	−0.270763	−	0.962646i	$-0.587276\pi$
$727$	−14.6011	−0.541526	−0.270763	+	0.962646i	$0.587276\pi$
$728$	0	0
$729$	−21.5260	−0.797260
$730$	0	0
$731$	−32.6951	−1.20927
$732$	0	0
$733$	−41.8369	−1.54528	−0.772640	−	0.634844i	$-0.781064\pi$
$733$	−41.8369	−1.54528	−0.772640	+	0.634844i	$0.781064\pi$
$734$	0	0
$735$	0.0679759	0.00250733
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.9299	−0.438849	−0.219424	−	0.975630i	$-0.570418\pi$
$739$	−11.9299	−0.438849	−0.219424	+	0.975630i	$0.570418\pi$
$740$	0	0
$741$	6.54559	0.240458
$742$	0	0
$743$	46.6803	1.71253	0.856267	−	0.516534i	$-0.172778\pi$
$743$	46.6803	1.71253	0.856267	+	0.516534i	$0.172778\pi$
$744$	0	0
$745$	14.6646	0.537270
$746$	0	0
$747$	−32.1873	−1.17767
$748$	0	0
$749$	48.4124	1.76895
$750$	0	0
$751$	−14.4039	−0.525607	−0.262803	−	0.964849i	$-0.584647\pi$
$751$	−14.4039	−0.525607	−0.262803	+	0.964849i	$0.584647\pi$
$752$	0	0
$753$	−2.01010	−0.0732523
$754$	0	0
$755$	7.51902	0.273645
$756$	0	0
$757$	−16.0616	−0.583768	−0.291884	−	0.956454i	$-0.594282\pi$
$757$	−16.0616	−0.583768	−0.291884	+	0.956454i	$0.594282\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.8840	1.40954	0.704772	−	0.709433i	$-0.251049\pi$
$761$	38.8840	1.40954	0.704772	+	0.709433i	$0.251049\pi$
$762$	0	0
$763$	43.9445	1.59090
$764$	0	0
$765$	13.3961	0.484337
$766$	0	0
$767$	−54.8345	−1.97996
$768$	0	0
$769$	−43.0017	−1.55068	−0.775341	−	0.631543i	$-0.782422\pi$
$769$	−43.0017	−1.55068	−0.775341	+	0.631543i	$0.782422\pi$
$770$	0	0
$771$	4.61679	0.166270
$772$	0	0
$773$	−7.85462	−0.282511	−0.141256	−	0.989973i	$-0.545114\pi$
$773$	−7.85462	−0.282511	−0.141256	+	0.989973i	$0.545114\pi$
$774$	0	0
$775$	−2.38630	−0.0857184
$776$	0	0
$777$	9.23290	0.331228
$778$	0	0
$779$	−9.59414	−0.343746
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.73967	0.205119
$784$	0	0
$785$	13.3819	0.477620
$786$	0	0
$787$	−11.4249	−0.407253	−0.203627	−	0.979049i	$-0.565273\pi$
$787$	−11.4249	−0.407253	−0.203627	+	0.979049i	$0.565273\pi$
$788$	0	0
$789$	−1.33471	−0.0475169
$790$	0	0
$791$	−5.51483	−0.196085
$792$	0	0
$793$	18.5903	0.660161
$794$	0	0
$795$	−2.04635	−0.0725764
$796$	0	0
$797$	−4.28502	−0.151783	−0.0758915	−	0.997116i	$-0.524180\pi$
$797$	−4.28502	−0.151783	−0.0758915	+	0.997116i	$0.524180\pi$
$798$	0	0
$799$	−20.1807	−0.713942
$800$	0	0
$801$	−8.02024	−0.283381
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7.44403	0.262368
$806$	0	0
$807$	0.544033	0.0191509
$808$	0	0
$809$	19.8484	0.697833	0.348917	−	0.937154i	$-0.386550\pi$
$809$	19.8484	0.697833	0.348917	+	0.937154i	$0.386550\pi$
$810$	0	0
$811$	−3.13836	−0.110203	−0.0551014	−	0.998481i	$-0.517548\pi$
$811$	−3.13836	−0.110203	−0.0551014	+	0.998481i	$0.517548\pi$
$812$	0	0
$813$	5.95452	0.208834
$814$	0	0
$815$	−0.771990	−0.0270416
$816$	0	0
$817$	30.6747	1.07317
$818$	0	0
$819$	−36.2943	−1.26823
$820$	0	0
$821$	−23.4999	−0.820151	−0.410076	−	0.912052i	$-0.634498\pi$
$821$	−23.4999	−0.820151	−0.410076	+	0.912052i	$0.634498\pi$
$822$	0	0
$823$	12.6564	0.441173	0.220586	−	0.975367i	$-0.429203\pi$
$823$	12.6564	0.441173	0.220586	+	0.975367i	$0.429203\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.2236	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$827$	30.2236	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$828$	0	0
$829$	2.00166	0.0695204	0.0347602	−	0.999396i	$-0.488933\pi$
$829$	2.00166	0.0695204	0.0347602	+	0.999396i	$0.488933\pi$
$830$	0	0
$831$	−1.10731	−0.0384121
$832$	0	0
$833$	0.973403	0.0337264
$834$	0	0
$835$	8.48232	0.293543
$836$	0	0
$837$	−4.54520	−0.157105
$838$	0	0
$839$	−35.3744	−1.22126	−0.610629	−	0.791917i	$-0.709083\pi$
$839$	−35.3744	−1.22126	−0.610629	+	0.791917i	$0.709083\pi$
$840$	0	0
$841$	−19.9193	−0.686873
$842$	0	0
$843$	−7.37304	−0.253941
$844$	0	0
$845$	−8.78880	−0.302344
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.40107	0.322644
$850$	0	0
$851$	29.5045	1.01140
$852$	0	0
$853$	18.2034	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$853$	18.2034	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$854$	0	0
$855$	−12.5683	−0.429827
$856$	0	0
$857$	29.2837	1.00031	0.500156	−	0.865935i	$-0.333276\pi$
$857$	29.2837	1.00031	0.500156	+	0.865935i	$0.333276\pi$
$858$	0	0
$859$	−8.44030	−0.287979	−0.143990	−	0.989579i	$-0.545993\pi$
$859$	−8.44030	−0.287979	−0.143990	+	0.989579i	$0.545993\pi$
$860$	0	0
$861$	−1.91756	−0.0653504
$862$	0	0
$863$	19.3487	0.658636	0.329318	−	0.944219i	$-0.393181\pi$
$863$	19.3487	0.658636	0.329318	+	0.944219i	$0.393181\pi$
$864$	0	0
$865$	5.07871	0.172681
$866$	0	0
$867$	−1.42244	−0.0483087
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−34.1679	−1.15773
$872$	0	0
$873$	−53.6681	−1.81639
$874$	0	0
$875$	−2.68522	−0.0907770
$876$	0	0
$877$	−17.5140	−0.591405	−0.295702	−	0.955280i	$-0.595554\pi$
$877$	−17.5140	−0.591405	−0.295702	+	0.955280i	$0.595554\pi$
$878$	0	0
$879$	6.85349	0.231163
$880$	0	0
$881$	−20.0575	−0.675754	−0.337877	−	0.941190i	$-0.609709\pi$
$881$	−20.0575	−0.675754	−0.337877	+	0.941190i	$0.609709\pi$
$882$	0	0
$883$	−26.8980	−0.905189	−0.452594	−	0.891717i	$-0.649501\pi$
$883$	−26.8980	−0.905189	−0.452594	+	0.891717i	$0.649501\pi$
$884$	0	0
$885$	−3.79521	−0.127574
$886$	0	0
$887$	6.80568	0.228512	0.114256	−	0.993451i	$-0.463552\pi$
$887$	6.80568	0.228512	0.114256	+	0.993451i	$0.463552\pi$
$888$	0	0
$889$	−0.204793	−0.00686854
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.9337	0.633591
$894$	0	0
$895$	−11.3170	−0.378287
$896$	0	0
$897$	4.18064	0.139588
$898$	0	0
$899$	−7.19091	−0.239830
$900$	0	0
$901$	−29.3033	−0.976235
$902$	0	0
$903$	6.13090	0.204024
$904$	0	0
$905$	−7.40006	−0.245986
$906$	0	0
$907$	21.3313	0.708295	0.354148	−	0.935190i	$-0.384771\pi$
$907$	21.3313	0.708295	0.354148	+	0.935190i	$0.384771\pi$
$908$	0	0
$909$	−20.6011	−0.683295
$910$	0	0
$911$	27.5888	0.914059	0.457029	−	0.889452i	$-0.348913\pi$
$911$	27.5888	0.914059	0.457029	+	0.889452i	$0.348913\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.28667	0.0425361
$916$	0	0
$917$	−30.8347	−1.01825
$918$	0	0
$919$	−6.00889	−0.198215	−0.0991075	−	0.995077i	$-0.531599\pi$
$919$	−6.00889	−0.198215	−0.0991075	+	0.995077i	$0.531599\pi$
$920$	0	0
$921$	2.22118	0.0731903
$922$	0	0
$923$	−5.58140	−0.183714
$924$	0	0
$925$	−10.6429	−0.349937
$926$	0	0
$927$	21.7993	0.715982
$928$	0	0
$929$	9.59739	0.314880	0.157440	−	0.987529i	$-0.449676\pi$
$929$	9.59739	0.314880	0.157440	+	0.987529i	$0.449676\pi$
$930$	0	0
$931$	−0.913252	−0.0299306
$932$	0	0
$933$	8.12771	0.266089
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.5917	1.26074	0.630369	−	0.776296i	$-0.282904\pi$
$937$	38.5917	1.26074	0.630369	+	0.776296i	$0.282904\pi$
$938$	0	0
$939$	3.74095	0.122081
$940$	0	0
$941$	29.1846	0.951391	0.475695	−	0.879610i	$-0.342196\pi$
$941$	29.1846	0.951391	0.475695	+	0.879610i	$0.342196\pi$
$942$	0	0
$943$	−6.12774	−0.199547
$944$	0	0
$945$	−5.11455	−0.166376
$946$	0	0
$947$	−46.7623	−1.51957	−0.759785	−	0.650174i	$-0.774696\pi$
$947$	−46.7623	−1.51957	−0.759785	+	0.650174i	$0.774696\pi$
$948$	0	0
$949$	−4.76917	−0.154814
$950$	0	0
$951$	−6.70090	−0.217292
$952$	0	0
$953$	6.15238	0.199295	0.0996475	−	0.995023i	$-0.468228\pi$
$953$	6.15238	0.199295	0.0996475	+	0.995023i	$0.468228\pi$
$954$	0	0
$955$	−5.15608	−0.166847
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−49.2182	−1.58934
$960$	0	0
$961$	−25.3056	−0.816309
$962$	0	0
$963$	−52.2058	−1.68231
$964$	0	0
$965$	−4.03230	−0.129804
$966$	0	0
$967$	3.39625	0.109216	0.0546080	−	0.998508i	$-0.482609\pi$
$967$	3.39625	0.109216	0.0546080	+	0.998508i	$0.482609\pi$
$968$	0	0
$969$	6.48736	0.208404
$970$	0	0
$971$	9.39500	0.301500	0.150750	−	0.988572i	$-0.451831\pi$
$971$	9.39500	0.301500	0.150750	+	0.988572i	$0.451831\pi$
$972$	0	0
$973$	−62.2631	−1.99606
$974$	0	0
$975$	−1.50805	−0.0482961
$976$	0	0
$977$	−16.5552	−0.529649	−0.264824	−	0.964297i	$-0.585314\pi$
$977$	−16.5552	−0.529649	−0.264824	+	0.964297i	$0.585314\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−47.3878	−1.51298
$982$	0	0
$983$	−50.8180	−1.62084	−0.810421	−	0.585848i	$-0.800762\pi$
$983$	−50.8180	−1.62084	−0.810421	+	0.585848i	$0.800762\pi$
$984$	0	0
$985$	11.4176	0.363794
$986$	0	0
$987$	3.78424	0.120454
$988$	0	0
$989$	19.5918	0.622983
$990$	0	0
$991$	−11.3642	−0.360996	−0.180498	−	0.983575i	$-0.557771\pi$
$991$	−11.3642	−0.360996	−0.180498	+	0.983575i	$0.557771\pi$
$992$	0	0
$993$	−10.3840	−0.329526
$994$	0	0
$995$	−7.16644	−0.227191
$996$	0	0
$997$	29.1644	0.923647	0.461824	−	0.886972i	$-0.347195\pi$
$997$	29.1644	0.923647	0.461824	+	0.886972i	$0.347195\pi$
$998$	0	0
$999$	−20.2716	−0.641365

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cm.1.3		4
4.3	odd	2		605.2.a.k.1.2		4
11.7	odd	10		880.2.bo.h.401.1		8
11.8	odd	10		880.2.bo.h.801.1		8
11.10	odd	2		9680.2.a.cn.1.3		4
12.11	even	2		5445.2.a.bi.1.3		4
20.19	odd	2		3025.2.a.w.1.3		4
44.3	odd	10		605.2.g.k.251.1		8
44.7	even	10		55.2.g.b.16.2	✓	8
44.15	odd	10		605.2.g.k.511.1		8
44.19	even	10		55.2.g.b.31.2	yes	8
44.27	odd	10		605.2.g.e.366.2		8
44.31	odd	10		605.2.g.e.81.2		8
44.35	even	10		605.2.g.m.81.1		8
44.39	even	10		605.2.g.m.366.1		8
44.43	even	2		605.2.a.j.1.3		4
132.95	odd	10		495.2.n.e.181.1		8
132.107	odd	10		495.2.n.e.361.1		8
132.131	odd	2		5445.2.a.bp.1.2		4
220.7	odd	20		275.2.z.a.49.2		16
220.19	even	10		275.2.h.a.251.1		8
220.63	odd	20		275.2.z.a.174.2		16
220.107	odd	20		275.2.z.a.174.3		16
220.139	even	10		275.2.h.a.126.1		8
220.183	odd	20		275.2.z.a.49.3		16
220.219	even	2		3025.2.a.bd.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.16.2	✓	8	44.7	even	10
55.2.g.b.31.2	yes	8	44.19	even	10
275.2.h.a.126.1		8	220.139	even	10
275.2.h.a.251.1		8	220.19	even	10
275.2.z.a.49.2		16	220.7	odd	20
275.2.z.a.49.3		16	220.183	odd	20
275.2.z.a.174.2		16	220.63	odd	20
275.2.z.a.174.3		16	220.107	odd	20
495.2.n.e.181.1		8	132.95	odd	10
495.2.n.e.361.1		8	132.107	odd	10
605.2.a.j.1.3		4	44.43	even	2
605.2.a.k.1.2		4	4.3	odd	2
605.2.g.e.81.2		8	44.31	odd	10
605.2.g.e.366.2		8	44.27	odd	10
605.2.g.k.251.1		8	44.3	odd	10
605.2.g.k.511.1		8	44.15	odd	10
605.2.g.m.81.1		8	44.35	even	10
605.2.g.m.366.1		8	44.39	even	10
880.2.bo.h.401.1		8	11.7	odd	10
880.2.bo.h.801.1		8	11.8	odd	10
3025.2.a.w.1.3		4	20.19	odd	2
3025.2.a.bd.1.2		4	220.219	even	2
5445.2.a.bi.1.3		4	12.11	even	2
5445.2.a.bp.1.2		4	132.131	odd	2
9680.2.a.cm.1.3		4	1.1	even	1	trivial
9680.2.a.cn.1.3		4	11.10	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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-0.323071 q^{3} -1.00000 q^{5} +2.68522 q^{7} -2.89563 q^{9} +4.66785 q^{13} +0.323071 q^{15} +4.62632 q^{17} -4.34044 q^{19} -0.867517 q^{21} -2.77222 q^{23} +1.00000 q^{25} +1.90471 q^{27} +3.01341 q^{29} -2.38630 q^{31} -2.68522 q^{35} -10.6429 q^{37} -1.50805 q^{39} +2.21041 q^{41} -7.06719 q^{43} +2.89563 q^{45} -4.36215 q^{47} +0.210405 q^{49} -1.49463 q^{51} -6.33404 q^{53} +1.40227 q^{57} -11.7473 q^{59} +3.98263 q^{61} -7.77539 q^{63} -4.66785 q^{65} -7.31984 q^{67} +0.895625 q^{69} -1.19571 q^{71} -1.02171 q^{73} -0.323071 q^{75} +3.50213 q^{79} +8.07152 q^{81} +11.1158 q^{83} -4.62632 q^{85} -0.973547 q^{87} +2.76978 q^{89} +12.5342 q^{91} +0.770945 q^{93} +4.34044 q^{95} +18.5342 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 3 q^{7} + q^{13} - q^{17} - 20 q^{19} + 10 q^{21} - 5 q^{23} + 4 q^{25} + 15 q^{27} + 12 q^{29} + 5 q^{31} + 3 q^{35} + 7 q^{37} - 7 q^{39} + 11 q^{41} - 19 q^{43} - 5 q^{47} + 3 q^{49}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 3 * q^7 + q^13 - q^17 - 20 * q^19 + 10 * q^21 - 5 * q^23 + 4 * q^25 + 15 * q^27 + 12 * q^29 + 5 * q^31 + 3 * q^35 + 7 * q^37 - 7 * q^39 + 11 * q^41 - 19 * q^43 - 5 * q^47 + 3 * q^49 - 7 * q^51 - 11 * q^53 - 5 * q^57 - 9 * q^59 + 12 * q^61 + 5 * q^63 - q^65 + 19 * q^67 - 8 * q^69 - 5 * q^71 + 11 * q^73 - 34 * q^79 + 4 * q^81 + 11 * q^83 + q^85 + 19 * q^87 - 8 * q^89 + 8 * q^91 - 5 * q^93 + 20 * q^95 + 32 * q^97

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