LMFDB - Embedded newform 9680.2.a.bm.1.1

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)

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);

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")

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 = [2,0,-1,0,-2,0,5,0,3,0,0,0,-6,0,1,0,3] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-2.56155 q^{3} -1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +1.12311 q^{13} +2.56155 q^{15} +7.68466 q^{17} +1.43845 q^{19} -11.6847 q^{21} -1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -8.56155 q^{29} +1.43845 q^{31} -4.56155 q^{35} +7.43845 q^{37} -2.87689 q^{39} +12.2462 q^{41} -3.12311 q^{43} -3.56155 q^{45} +11.3693 q^{47} +13.8078 q^{49} -19.6847 q^{51} +9.68466 q^{53} -3.68466 q^{57} -1.12311 q^{59} +12.5616 q^{61} +16.2462 q^{63} -1.12311 q^{65} +2.87689 q^{69} -3.68466 q^{71} -1.12311 q^{73} -2.56155 q^{75} -11.3693 q^{79} -7.00000 q^{81} -6.00000 q^{83} -7.68466 q^{85} +21.9309 q^{87} +9.68466 q^{89} +5.12311 q^{91} -3.68466 q^{93} -1.43845 q^{95} -4.87689 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 6 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} - 11 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} - 14 q^{39} + 8 q^{41} + 2 q^{43}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 6 * q^13 + q^15 + 3 * q^17 + 7 * q^19 - 11 * q^21 + 6 * q^23 + 2 * q^25 - 7 * q^27 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 - 14 * q^39 + 8 * q^41 + 2 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 - 27 * q^51 + 7 * q^53 + 5 * q^57 + 6 * q^59 + 21 * q^61 + 16 * q^63 + 6 * q^65 + 14 * q^69 + 5 * q^71 + 6 * q^73 - q^75 + 2 * q^79 - 14 * q^81 - 12 * q^83 - 3 * q^85 + 15 * q^87 + 7 * q^89 + 2 * q^91 + 5 * q^93 - 7 * q^95 - 18 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.56155	1.72410	0.862052	−	0.506819i	$-0.169179\pi$
$7$	4.56155	1.72410	0.862052	+	0.506819i	$0.169179\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.12311	0.311493	0.155747	−	0.987797i	$-0.450222\pi$
$13$	1.12311	0.311493	0.155747	+	0.987797i	$0.450222\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	7.68466	1.86380	0.931902	−	0.362711i	$-0.118149\pi$
$17$	7.68466	1.86380	0.931902	+	0.362711i	$0.118149\pi$
$18$	0	0
$19$	1.43845	0.330002	0.165001	−	0.986293i	$-0.447237\pi$
$19$	1.43845	0.330002	0.165001	+	0.986293i	$0.447237\pi$
$20$	0	0
$21$	−11.6847	−2.54980
$22$	0	0
$23$	−1.12311	−0.234184	−0.117092	−	0.993121i	$-0.537357\pi$
$23$	−1.12311	−0.234184	−0.117092	+	0.993121i	$0.537357\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	−8.56155	−1.58984	−0.794920	−	0.606714i	$-0.792487\pi$
$29$	−8.56155	−1.58984	−0.794920	+	0.606714i	$0.792487\pi$
$30$	0	0
$31$	1.43845	0.258353	0.129176	−	0.991622i	$-0.458767\pi$
$31$	1.43845	0.258353	0.129176	+	0.991622i	$0.458767\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.56155	−0.771043
$36$	0	0
$37$	7.43845	1.22287	0.611437	−	0.791293i	$-0.290592\pi$
$37$	7.43845	1.22287	0.611437	+	0.791293i	$0.290592\pi$
$38$	0	0
$39$	−2.87689	−0.460672
$40$	0	0
$41$	12.2462	1.91254	0.956268	−	0.292490i	$-0.0944840\pi$
$41$	12.2462	1.91254	0.956268	+	0.292490i	$0.0944840\pi$
$42$	0	0
$43$	−3.12311	−0.476269	−0.238135	−	0.971232i	$-0.576536\pi$
$43$	−3.12311	−0.476269	−0.238135	+	0.971232i	$0.576536\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	11.3693	1.65839	0.829193	−	0.558963i	$-0.188801\pi$
$47$	11.3693	1.65839	0.829193	+	0.558963i	$0.188801\pi$
$48$	0	0
$49$	13.8078	1.97254
$50$	0	0
$51$	−19.6847	−2.75640
$52$	0	0
$53$	9.68466	1.33029	0.665145	−	0.746714i	$-0.268370\pi$
$53$	9.68466	1.33029	0.665145	+	0.746714i	$0.268370\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.68466	−0.488045
$58$	0	0
$59$	−1.12311	−0.146216	−0.0731079	−	0.997324i	$-0.523292\pi$
$59$	−1.12311	−0.146216	−0.0731079	+	0.997324i	$0.523292\pi$
$60$	0	0
$61$	12.5616	1.60834	0.804171	−	0.594398i	$-0.202610\pi$
$61$	12.5616	1.60834	0.804171	+	0.594398i	$0.202610\pi$
$62$	0	0
$63$	16.2462	2.04683
$64$	0	0
$65$	−1.12311	−0.139304
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	2.87689	0.346337
$70$	0	0
$71$	−3.68466	−0.437289	−0.218644	−	0.975805i	$-0.570163\pi$
$71$	−3.68466	−0.437289	−0.218644	+	0.975805i	$0.570163\pi$
$72$	0	0
$73$	−1.12311	−0.131450	−0.0657248	−	0.997838i	$-0.520936\pi$
$73$	−1.12311	−0.131450	−0.0657248	+	0.997838i	$0.520936\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.3693	−1.27915	−0.639574	−	0.768729i	$-0.720889\pi$
$79$	−11.3693	−1.27915	−0.639574	+	0.768729i	$0.720889\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−7.68466	−0.833518
$86$	0	0
$87$	21.9309	2.35124
$88$	0	0
$89$	9.68466	1.02657	0.513286	−	0.858218i	$-0.328428\pi$
$89$	9.68466	1.02657	0.513286	+	0.858218i	$0.328428\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	0	0
$93$	−3.68466	−0.382081
$94$	0	0
$95$	−1.43845	−0.147582
$96$	0	0
$97$	−4.87689	−0.495174	−0.247587	−	0.968866i	$-0.579638\pi$
$97$	−4.87689	−0.495174	−0.247587	+	0.968866i	$0.579638\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	11.6847	1.14031
$106$	0	0
$107$	7.12311	0.688617	0.344308	−	0.938857i	$-0.388113\pi$
$107$	7.12311	0.688617	0.344308	+	0.938857i	$0.388113\pi$
$108$	0	0
$109$	4.24621	0.406713	0.203357	−	0.979105i	$-0.434815\pi$
$109$	4.24621	0.406713	0.203357	+	0.979105i	$0.434815\pi$
$110$	0	0
$111$	−19.0540	−1.80852
$112$	0	0
$113$	0.876894	0.0824913	0.0412456	−	0.999149i	$-0.486867\pi$
$113$	0.876894	0.0824913	0.0412456	+	0.999149i	$0.486867\pi$
$114$	0	0
$115$	1.12311	0.104730
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	35.0540	3.21339
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−31.3693	−2.82848
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.87689	0.432754	0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689	0.432754	0.216377	+	0.976310i	$0.430576\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−16.8078	−1.46850	−0.734251	−	0.678879i	$-0.762466\pi$
$131$	−16.8078	−1.46850	−0.734251	+	0.678879i	$0.762466\pi$
$132$	0	0
$133$	6.56155	0.568959
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−7.12311	−0.608568	−0.304284	−	0.952581i	$-0.598417\pi$
$137$	−7.12311	−0.608568	−0.304284	+	0.952581i	$0.598417\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−29.1231	−2.45261
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.56155	0.710998
$146$	0	0
$147$	−35.3693	−2.91721
$148$	0	0
$149$	−1.68466	−0.138013	−0.0690063	−	0.997616i	$-0.521983\pi$
$149$	−1.68466	−0.138013	−0.0690063	+	0.997616i	$0.521983\pi$
$150$	0	0
$151$	6.87689	0.559634	0.279817	−	0.960053i	$-0.409726\pi$
$151$	6.87689	0.559634	0.279817	+	0.960053i	$0.409726\pi$
$152$	0	0
$153$	27.3693	2.21268
$154$	0	0
$155$	−1.43845	−0.115539
$156$	0	0
$157$	21.6847	1.73062	0.865312	−	0.501233i	$-0.167120\pi$
$157$	21.6847	1.73062	0.865312	+	0.501233i	$0.167120\pi$
$158$	0	0
$159$	−24.8078	−1.96738
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	−20.8078	−1.62979	−0.814895	−	0.579609i	$-0.803205\pi$
$163$	−20.8078	−1.62979	−0.814895	+	0.579609i	$0.803205\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.68466	−0.130363	−0.0651814	−	0.997873i	$-0.520763\pi$
$167$	−1.68466	−0.130363	−0.0651814	+	0.997873i	$0.520763\pi$
$168$	0	0
$169$	−11.7386	−0.902972
$170$	0	0
$171$	5.12311	0.391774
$172$	0	0
$173$	−10.8769	−0.826955	−0.413477	−	0.910514i	$-0.635686\pi$
$173$	−10.8769	−0.826955	−0.413477	+	0.910514i	$0.635686\pi$
$174$	0	0
$175$	4.56155	0.344821
$176$	0	0
$177$	2.87689	0.216241
$178$	0	0
$179$	−14.2462	−1.06481	−0.532406	−	0.846489i	$-0.678712\pi$
$179$	−14.2462	−1.06481	−0.532406	+	0.846489i	$0.678712\pi$
$180$	0	0
$181$	−8.24621	−0.612936	−0.306468	−	0.951881i	$-0.599147\pi$
$181$	−8.24621	−0.612936	−0.306468	+	0.951881i	$0.599147\pi$
$182$	0	0
$183$	−32.1771	−2.37860
$184$	0	0
$185$	−7.43845	−0.546886
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.56155	−0.477283
$190$	0	0
$191$	−10.2462	−0.741390	−0.370695	−	0.928755i	$-0.620880\pi$
$191$	−10.2462	−0.741390	−0.370695	+	0.928755i	$0.620880\pi$
$192$	0	0
$193$	21.4384	1.54317	0.771587	−	0.636124i	$-0.219463\pi$
$193$	21.4384	1.54317	0.771587	+	0.636124i	$0.219463\pi$
$194$	0	0
$195$	2.87689	0.206019
$196$	0	0
$197$	−22.7386	−1.62006	−0.810030	−	0.586388i	$-0.800549\pi$
$197$	−22.7386	−1.62006	−0.810030	+	0.586388i	$0.800549\pi$
$198$	0	0
$199$	4.31534	0.305906	0.152953	−	0.988233i	$-0.451122\pi$
$199$	4.31534	0.305906	0.152953	+	0.988233i	$0.451122\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−39.0540	−2.74105
$204$	0	0
$205$	−12.2462	−0.855312
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.6847	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$211$	19.6847	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$212$	0	0
$213$	9.43845	0.646712
$214$	0	0
$215$	3.12311	0.212994
$216$	0	0
$217$	6.56155	0.445427
$218$	0	0
$219$	2.87689	0.194403
$220$	0	0
$221$	8.63068	0.580563
$222$	0	0
$223$	−25.1231	−1.68237	−0.841184	−	0.540749i	$-0.818141\pi$
$223$	−25.1231	−1.68237	−0.841184	+	0.540749i	$0.818141\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	8.24621	0.547320	0.273660	−	0.961826i	$-0.411766\pi$
$227$	8.24621	0.547320	0.273660	+	0.961826i	$0.411766\pi$
$228$	0	0
$229$	16.2462	1.07358	0.536790	−	0.843716i	$-0.319637\pi$
$229$	16.2462	1.07358	0.536790	+	0.843716i	$0.319637\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.0540	−1.77236	−0.886182	−	0.463336i	$-0.846652\pi$
$233$	−27.0540	−1.77236	−0.886182	+	0.463336i	$0.846652\pi$
$234$	0	0
$235$	−11.3693	−0.741652
$236$	0	0
$237$	29.1231	1.89175
$238$	0	0
$239$	−21.6155	−1.39819	−0.699096	−	0.715028i	$-0.746414\pi$
$239$	−21.6155	−1.39819	−0.699096	+	0.715028i	$0.746414\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	−13.8078	−0.882146
$246$	0	0
$247$	1.61553	0.102794
$248$	0	0
$249$	15.3693	0.973991
$250$	0	0
$251$	27.3693	1.72754	0.863768	−	0.503890i	$-0.168098\pi$
$251$	27.3693	1.72754	0.863768	+	0.503890i	$0.168098\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	19.6847	1.23270
$256$	0	0
$257$	6.49242	0.404986	0.202493	−	0.979284i	$-0.435096\pi$
$257$	6.49242	0.404986	0.202493	+	0.979284i	$0.435096\pi$
$258$	0	0
$259$	33.9309	2.10836
$260$	0	0
$261$	−30.4924	−1.88743
$262$	0	0
$263$	−29.0540	−1.79154	−0.895772	−	0.444513i	$-0.853377\pi$
$263$	−29.0540	−1.79154	−0.895772	+	0.444513i	$0.853377\pi$
$264$	0	0
$265$	−9.68466	−0.594924
$266$	0	0
$267$	−24.8078	−1.51821
$268$	0	0
$269$	0.876894	0.0534652	0.0267326	−	0.999643i	$-0.491490\pi$
$269$	0.876894	0.0534652	0.0267326	+	0.999643i	$0.491490\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	−13.1231	−0.794246
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.2462	1.33665	0.668323	−	0.743872i	$-0.267013\pi$
$277$	22.2462	1.33665	0.668323	+	0.743872i	$0.267013\pi$
$278$	0	0
$279$	5.12311	0.306712
$280$	0	0
$281$	−8.24621	−0.491928	−0.245964	−	0.969279i	$-0.579104\pi$
$281$	−8.24621	−0.491928	−0.245964	+	0.969279i	$0.579104\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	3.68466	0.218260
$286$	0	0
$287$	55.8617	3.29741
$288$	0	0
$289$	42.0540	2.47376
$290$	0	0
$291$	12.4924	0.732319
$292$	0	0
$293$	−17.1231	−1.00034	−0.500171	−	0.865927i	$-0.666730\pi$
$293$	−17.1231	−1.00034	−0.500171	+	0.865927i	$0.666730\pi$
$294$	0	0
$295$	1.12311	0.0653897
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.26137	−0.0729467
$300$	0	0
$301$	−14.2462	−0.821138
$302$	0	0
$303$	−5.12311	−0.294315
$304$	0	0
$305$	−12.5616	−0.719272
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.43845	0.535205	0.267603	−	0.963529i	$-0.413769\pi$
$311$	9.43845	0.535205	0.267603	+	0.963529i	$0.413769\pi$
$312$	0	0
$313$	24.7386	1.39831	0.699155	−	0.714970i	$-0.253560\pi$
$313$	24.7386	1.39831	0.699155	+	0.714970i	$0.253560\pi$
$314$	0	0
$315$	−16.2462	−0.915370
$316$	0	0
$317$	6.31534	0.354705	0.177352	−	0.984147i	$-0.443247\pi$
$317$	6.31534	0.354705	0.177352	+	0.984147i	$0.443247\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.2462	−1.01840
$322$	0	0
$323$	11.0540	0.615060
$324$	0	0
$325$	1.12311	0.0622987
$326$	0	0
$327$	−10.8769	−0.601494
$328$	0	0
$329$	51.8617	2.85923
$330$	0	0
$331$	18.7386	1.02997	0.514984	−	0.857200i	$-0.327798\pi$
$331$	18.7386	1.02997	0.514984	+	0.857200i	$0.327798\pi$
$332$	0	0
$333$	26.4924	1.45178
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.80776	0.261896	0.130948	−	0.991389i	$-0.458198\pi$
$337$	4.80776	0.261896	0.130948	+	0.991389i	$0.458198\pi$
$338$	0	0
$339$	−2.24621	−0.121997
$340$	0	0
$341$	0	0
$342$	0	0
$343$	31.0540	1.67676
$344$	0	0
$345$	−2.87689	−0.154887
$346$	0	0
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	−14.4924	−0.775762	−0.387881	−	0.921710i	$-0.626793\pi$
$349$	−14.4924	−0.775762	−0.387881	+	0.921710i	$0.626793\pi$
$350$	0	0
$351$	−1.61553	−0.0862305
$352$	0	0
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	3.68466	0.195561
$356$	0	0
$357$	−89.7926	−4.75233
$358$	0	0
$359$	18.7386	0.988987	0.494494	−	0.869181i	$-0.335354\pi$
$359$	18.7386	0.988987	0.494494	+	0.869181i	$0.335354\pi$
$360$	0	0
$361$	−16.9309	−0.891098
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.12311	0.0587860
$366$	0	0
$367$	−16.4924	−0.860897	−0.430449	−	0.902615i	$-0.641645\pi$
$367$	−16.4924	−0.860897	−0.430449	+	0.902615i	$0.641645\pi$
$368$	0	0
$369$	43.6155	2.27053
$370$	0	0
$371$	44.1771	2.29356
$372$	0	0
$373$	−23.3693	−1.21002	−0.605009	−	0.796219i	$-0.706830\pi$
$373$	−23.3693	−1.21002	−0.605009	+	0.796219i	$0.706830\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	−9.61553	−0.495225
$378$	0	0
$379$	14.8769	0.764175	0.382087	−	0.924126i	$-0.375205\pi$
$379$	14.8769	0.764175	0.382087	+	0.924126i	$0.375205\pi$
$380$	0	0
$381$	−12.4924	−0.640006
$382$	0	0
$383$	0.630683	0.0322264	0.0161132	−	0.999870i	$-0.494871\pi$
$383$	0.630683	0.0322264	0.0161132	+	0.999870i	$0.494871\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.1231	−0.565419
$388$	0	0
$389$	−14.4924	−0.734795	−0.367397	−	0.930064i	$-0.619751\pi$
$389$	−14.4924	−0.734795	−0.367397	+	0.930064i	$0.619751\pi$
$390$	0	0
$391$	−8.63068	−0.436472
$392$	0	0
$393$	43.0540	2.17179
$394$	0	0
$395$	11.3693	0.572052
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	−16.8078	−0.841441
$400$	0	0
$401$	30.8078	1.53847	0.769233	−	0.638968i	$-0.220638\pi$
$401$	30.8078	1.53847	0.769233	+	0.638968i	$0.220638\pi$
$402$	0	0
$403$	1.61553	0.0804752
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−17.3693	−0.858857	−0.429429	−	0.903101i	$-0.641285\pi$
$409$	−17.3693	−0.858857	−0.429429	+	0.903101i	$0.641285\pi$
$410$	0	0
$411$	18.2462	0.900019
$412$	0	0
$413$	−5.12311	−0.252092
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	10.2462	0.501759
$418$	0	0
$419$	−0.492423	−0.0240564	−0.0120282	−	0.999928i	$-0.503829\pi$
$419$	−0.492423	−0.0240564	−0.0120282	+	0.999928i	$0.503829\pi$
$420$	0	0
$421$	27.6155	1.34590	0.672949	−	0.739689i	$-0.265027\pi$
$421$	27.6155	1.34590	0.672949	+	0.739689i	$0.265027\pi$
$422$	0	0
$423$	40.4924	1.96881
$424$	0	0
$425$	7.68466	0.372761
$426$	0	0
$427$	57.3002	2.77295
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−4.87689	−0.234369	−0.117184	−	0.993110i	$-0.537387\pi$
$433$	−4.87689	−0.234369	−0.117184	+	0.993110i	$0.537387\pi$
$434$	0	0
$435$	−21.9309	−1.05150
$436$	0	0
$437$	−1.61553	−0.0772812
$438$	0	0
$439$	10.2462	0.489025	0.244512	−	0.969646i	$-0.421372\pi$
$439$	10.2462	0.489025	0.244512	+	0.969646i	$0.421372\pi$
$440$	0	0
$441$	49.1771	2.34177
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−9.68466	−0.459097
$446$	0	0
$447$	4.31534	0.204109
$448$	0	0
$449$	0.246211	0.0116194	0.00580971	−	0.999983i	$-0.498151\pi$
$449$	0.246211	0.0116194	0.00580971	+	0.999983i	$0.498151\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−17.6155	−0.827650
$454$	0	0
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−3.68466	−0.172361	−0.0861805	−	0.996280i	$-0.527466\pi$
$457$	−3.68466	−0.172361	−0.0861805	+	0.996280i	$0.527466\pi$
$458$	0	0
$459$	−11.0540	−0.515955
$460$	0	0
$461$	21.0540	0.980581	0.490291	−	0.871559i	$-0.336891\pi$
$461$	21.0540	0.980581	0.490291	+	0.871559i	$0.336891\pi$
$462$	0	0
$463$	27.3693	1.27196	0.635980	−	0.771706i	$-0.280596\pi$
$463$	27.3693	1.27196	0.635980	+	0.771706i	$0.280596\pi$
$464$	0	0
$465$	3.68466	0.170872
$466$	0	0
$467$	−32.8078	−1.51816	−0.759081	−	0.650996i	$-0.774351\pi$
$467$	−32.8078	−1.51816	−0.759081	+	0.650996i	$0.774351\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−55.5464	−2.55944
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.43845	0.0660005
$476$	0	0
$477$	34.4924	1.57930
$478$	0	0
$479$	14.2462	0.650926	0.325463	−	0.945555i	$-0.394480\pi$
$479$	14.2462	0.650926	0.325463	+	0.945555i	$0.394480\pi$
$480$	0	0
$481$	8.35416	0.380917
$482$	0	0
$483$	13.1231	0.597122
$484$	0	0
$485$	4.87689	0.221448
$486$	0	0
$487$	30.2462	1.37059	0.685293	−	0.728267i	$-0.259674\pi$
$487$	30.2462	1.37059	0.685293	+	0.728267i	$0.259674\pi$
$488$	0	0
$489$	53.3002	2.41032
$490$	0	0
$491$	−16.3153	−0.736301	−0.368151	−	0.929766i	$-0.620009\pi$
$491$	−16.3153	−0.736301	−0.368151	+	0.929766i	$0.620009\pi$
$492$	0	0
$493$	−65.7926	−2.96315
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.8078	−0.753931
$498$	0	0
$499$	18.7386	0.838856	0.419428	−	0.907789i	$-0.362231\pi$
$499$	18.7386	0.838856	0.419428	+	0.907789i	$0.362231\pi$
$500$	0	0
$501$	4.31534	0.192795
$502$	0	0
$503$	−3.12311	−0.139252	−0.0696262	−	0.997573i	$-0.522181\pi$
$503$	−3.12311	−0.139252	−0.0696262	+	0.997573i	$0.522181\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	30.0691	1.33542
$508$	0	0
$509$	−0.384472	−0.0170414	−0.00852071	−	0.999964i	$-0.502712\pi$
$509$	−0.384472	−0.0170414	−0.00852071	+	0.999964i	$0.502712\pi$
$510$	0	0
$511$	−5.12311	−0.226633
$512$	0	0
$513$	−2.06913	−0.0913543
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	27.8617	1.22299
$520$	0	0
$521$	−30.4924	−1.33590	−0.667949	−	0.744207i	$-0.732827\pi$
$521$	−30.4924	−1.33590	−0.667949	+	0.744207i	$0.732827\pi$
$522$	0	0
$523$	24.8769	1.08779	0.543895	−	0.839153i	$-0.316949\pi$
$523$	24.8769	1.08779	0.543895	+	0.839153i	$0.316949\pi$
$524$	0	0
$525$	−11.6847	−0.509960
$526$	0	0
$527$	11.0540	0.481519
$528$	0	0
$529$	−21.7386	−0.945158
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	13.7538	0.595743
$534$	0	0
$535$	−7.12311	−0.307959
$536$	0	0
$537$	36.4924	1.57476
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.93087	0.340975	0.170487	−	0.985360i	$-0.445466\pi$
$541$	7.93087	0.340975	0.170487	+	0.985360i	$0.445466\pi$
$542$	0	0
$543$	21.1231	0.906479
$544$	0	0
$545$	−4.24621	−0.181888
$546$	0	0
$547$	−10.6307	−0.454535	−0.227268	−	0.973832i	$-0.572979\pi$
$547$	−10.6307	−0.454535	−0.227268	+	0.973832i	$0.572979\pi$
$548$	0	0
$549$	44.7386	1.90940
$550$	0	0
$551$	−12.3153	−0.524651
$552$	0	0
$553$	−51.8617	−2.20539
$554$	0	0
$555$	19.0540	0.808796
$556$	0	0
$557$	−16.6307	−0.704665	−0.352332	−	0.935875i	$-0.614611\pi$
$557$	−16.6307	−0.704665	−0.352332	+	0.935875i	$0.614611\pi$
$558$	0	0
$559$	−3.50758	−0.148355
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.9848	−1.81160	−0.905798	−	0.423711i	$-0.860727\pi$
$563$	−42.9848	−1.81160	−0.905798	+	0.423711i	$0.860727\pi$
$564$	0	0
$565$	−0.876894	−0.0368912
$566$	0	0
$567$	−31.9309	−1.34097
$568$	0	0
$569$	−3.61553	−0.151571	−0.0757854	−	0.997124i	$-0.524146\pi$
$569$	−3.61553	−0.151571	−0.0757854	+	0.997124i	$0.524146\pi$
$570$	0	0
$571$	25.3002	1.05878	0.529390	−	0.848379i	$-0.322421\pi$
$571$	25.3002	1.05878	0.529390	+	0.848379i	$0.322421\pi$
$572$	0	0
$573$	26.2462	1.09645
$574$	0	0
$575$	−1.12311	−0.0468367
$576$	0	0
$577$	−15.7538	−0.655839	−0.327919	−	0.944706i	$-0.606347\pi$
$577$	−15.7538	−0.655839	−0.327919	+	0.944706i	$0.606347\pi$
$578$	0	0
$579$	−54.9157	−2.28222
$580$	0	0
$581$	−27.3693	−1.13547
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	3.05398	0.126051	0.0630255	−	0.998012i	$-0.479925\pi$
$587$	3.05398	0.126051	0.0630255	+	0.998012i	$0.479925\pi$
$588$	0	0
$589$	2.06913	0.0852570
$590$	0	0
$591$	58.2462	2.39593
$592$	0	0
$593$	−21.6155	−0.887643	−0.443822	−	0.896115i	$-0.646378\pi$
$593$	−21.6155	−0.887643	−0.443822	+	0.896115i	$0.646378\pi$
$594$	0	0
$595$	−35.0540	−1.43707
$596$	0	0
$597$	−11.0540	−0.452409
$598$	0	0
$599$	29.9309	1.22294	0.611471	−	0.791267i	$-0.290578\pi$
$599$	29.9309	1.22294	0.611471	+	0.791267i	$0.290578\pi$
$600$	0	0
$601$	26.9848	1.10073	0.550367	−	0.834923i	$-0.314488\pi$
$601$	26.9848	1.10073	0.550367	+	0.834923i	$0.314488\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.8078	−0.601029	−0.300514	−	0.953777i	$-0.597158\pi$
$607$	−14.8078	−0.601029	−0.300514	+	0.953777i	$0.597158\pi$
$608$	0	0
$609$	100.039	4.05378
$610$	0	0
$611$	12.7689	0.516576
$612$	0	0
$613$	2.87689	0.116197	0.0580983	−	0.998311i	$-0.481496\pi$
$613$	2.87689	0.116197	0.0580983	+	0.998311i	$0.481496\pi$
$614$	0	0
$615$	31.3693	1.26493
$616$	0	0
$617$	−27.6155	−1.11176	−0.555880	−	0.831263i	$-0.687618\pi$
$617$	−27.6155	−1.11176	−0.555880	+	0.831263i	$0.687618\pi$
$618$	0	0
$619$	34.7386	1.39626	0.698132	−	0.715969i	$-0.254015\pi$
$619$	34.7386	1.39626	0.698132	+	0.715969i	$0.254015\pi$
$620$	0	0
$621$	1.61553	0.0648289
$622$	0	0
$623$	44.1771	1.76992
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	57.1619	2.27920
$630$	0	0
$631$	32.8078	1.30606	0.653028	−	0.757334i	$-0.273498\pi$
$631$	32.8078	1.30606	0.653028	+	0.757334i	$0.273498\pi$
$632$	0	0
$633$	−50.4233	−2.00415
$634$	0	0
$635$	−4.87689	−0.193534
$636$	0	0
$637$	15.5076	0.614433
$638$	0	0
$639$	−13.1231	−0.519142
$640$	0	0
$641$	20.5616	0.812133	0.406066	−	0.913844i	$-0.366900\pi$
$641$	20.5616	0.812133	0.406066	+	0.913844i	$0.366900\pi$
$642$	0	0
$643$	−25.3002	−0.997742	−0.498871	−	0.866676i	$-0.666252\pi$
$643$	−25.3002	−0.997742	−0.498871	+	0.866676i	$0.666252\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−45.1231	−1.77397	−0.886986	−	0.461796i	$-0.847205\pi$
$647$	−45.1231	−1.77397	−0.886986	+	0.461796i	$0.847205\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.8078	−0.658748
$652$	0	0
$653$	28.5616	1.11770	0.558850	−	0.829269i	$-0.311243\pi$
$653$	28.5616	1.11770	0.558850	+	0.829269i	$0.311243\pi$
$654$	0	0
$655$	16.8078	0.656734
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−33.9309	−1.32176	−0.660880	−	0.750492i	$-0.729817\pi$
$659$	−33.9309	−1.32176	−0.660880	+	0.750492i	$0.729817\pi$
$660$	0	0
$661$	2.49242	0.0969440	0.0484720	−	0.998825i	$-0.484565\pi$
$661$	2.49242	0.0969440	0.0484720	+	0.998825i	$0.484565\pi$
$662$	0	0
$663$	−22.1080	−0.858602
$664$	0	0
$665$	−6.56155	−0.254446
$666$	0	0
$667$	9.61553	0.372315
$668$	0	0
$669$	64.3542	2.48808
$670$	0	0
$671$	0	0
$672$	0	0
$673$	41.9309	1.61632	0.808158	−	0.588966i	$-0.200465\pi$
$673$	41.9309	1.61632	0.808158	+	0.588966i	$0.200465\pi$
$674$	0	0
$675$	−1.43845	−0.0553659
$676$	0	0
$677$	−46.2462	−1.77739	−0.888693	−	0.458502i	$-0.848386\pi$
$677$	−46.2462	−1.77739	−0.888693	+	0.458502i	$0.848386\pi$
$678$	0	0
$679$	−22.2462	−0.853731
$680$	0	0
$681$	−21.1231	−0.809439
$682$	0	0
$683$	−40.1771	−1.53733	−0.768667	−	0.639650i	$-0.779079\pi$
$683$	−40.1771	−1.53733	−0.768667	+	0.639650i	$0.779079\pi$
$684$	0	0
$685$	7.12311	0.272160
$686$	0	0
$687$	−41.6155	−1.58773
$688$	0	0
$689$	10.8769	0.414377
$690$	0	0
$691$	−5.61553	−0.213625	−0.106812	−	0.994279i	$-0.534064\pi$
$691$	−5.61553	−0.213625	−0.106812	+	0.994279i	$0.534064\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	94.1080	3.56459
$698$	0	0
$699$	69.3002	2.62117
$700$	0	0
$701$	20.5616	0.776599	0.388300	−	0.921533i	$-0.373063\pi$
$701$	20.5616	0.776599	0.388300	+	0.921533i	$0.373063\pi$
$702$	0	0
$703$	10.6998	0.403551
$704$	0	0
$705$	29.1231	1.09684
$706$	0	0
$707$	9.12311	0.343110
$708$	0	0
$709$	−32.2462	−1.21103	−0.605516	−	0.795833i	$-0.707033\pi$
$709$	−32.2462	−1.21103	−0.605516	+	0.795833i	$0.707033\pi$
$710$	0	0
$711$	−40.4924	−1.51858
$712$	0	0
$713$	−1.61553	−0.0605020
$714$	0	0
$715$	0	0
$716$	0	0
$717$	55.3693	2.06781
$718$	0	0
$719$	48.1771	1.79670	0.898351	−	0.439278i	$-0.144766\pi$
$719$	48.1771	1.79670	0.898351	+	0.439278i	$0.144766\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	47.3693	1.76168
$724$	0	0
$725$	−8.56155	−0.317968
$726$	0	0
$727$	10.8769	0.403402	0.201701	−	0.979447i	$-0.435353\pi$
$727$	10.8769	0.403402	0.201701	+	0.979447i	$0.435353\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	13.7538	0.508008	0.254004	−	0.967203i	$-0.418252\pi$
$733$	13.7538	0.508008	0.254004	+	0.967203i	$0.418252\pi$
$734$	0	0
$735$	35.3693	1.30462
$736$	0	0
$737$	0	0
$738$	0	0
$739$	22.2462	0.818340	0.409170	−	0.912458i	$-0.365818\pi$
$739$	22.2462	0.818340	0.409170	+	0.912458i	$0.365818\pi$
$740$	0	0
$741$	−4.13826	−0.152023
$742$	0	0
$743$	20.5616	0.754330	0.377165	−	0.926146i	$-0.376899\pi$
$743$	20.5616	0.754330	0.377165	+	0.926146i	$0.376899\pi$
$744$	0	0
$745$	1.68466	0.0617211
$746$	0	0
$747$	−21.3693	−0.781862
$748$	0	0
$749$	32.4924	1.18725
$750$	0	0
$751$	31.1922	1.13822	0.569110	−	0.822261i	$-0.307288\pi$
$751$	31.1922	1.13822	0.569110	+	0.822261i	$0.307288\pi$
$752$	0	0
$753$	−70.1080	−2.55488
$754$	0	0
$755$	−6.87689	−0.250276
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.75379	−0.281075	−0.140537	−	0.990075i	$-0.544883\pi$
$761$	−7.75379	−0.281075	−0.140537	+	0.990075i	$0.544883\pi$
$762$	0	0
$763$	19.3693	0.701216
$764$	0	0
$765$	−27.3693	−0.989540
$766$	0	0
$767$	−1.26137	−0.0455453
$768$	0	0
$769$	−3.75379	−0.135365	−0.0676825	−	0.997707i	$-0.521561\pi$
$769$	−3.75379	−0.135365	−0.0676825	+	0.997707i	$0.521561\pi$
$770$	0	0
$771$	−16.6307	−0.598939
$772$	0	0
$773$	43.9309	1.58008	0.790042	−	0.613053i	$-0.210059\pi$
$773$	43.9309	1.58008	0.790042	+	0.613053i	$0.210059\pi$
$774$	0	0
$775$	1.43845	0.0516705
$776$	0	0
$777$	−86.9157	−3.11808
$778$	0	0
$779$	17.6155	0.631142
$780$	0	0
$781$	0	0
$782$	0	0
$783$	12.3153	0.440114
$784$	0	0
$785$	−21.6847	−0.773959
$786$	0	0
$787$	−17.8617	−0.636702	−0.318351	−	0.947973i	$-0.603129\pi$
$787$	−17.8617	−0.636702	−0.318351	+	0.947973i	$0.603129\pi$
$788$	0	0
$789$	74.4233	2.64954
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	14.1080	0.500988
$794$	0	0
$795$	24.8078	0.879841
$796$	0	0
$797$	20.2462	0.717158	0.358579	−	0.933499i	$-0.383261\pi$
$797$	20.2462	0.717158	0.358579	+	0.933499i	$0.383261\pi$
$798$	0	0
$799$	87.3693	3.09090
$800$	0	0
$801$	34.4924	1.21873
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.12311	0.180566
$806$	0	0
$807$	−2.24621	−0.0790704
$808$	0	0
$809$	40.7386	1.43229	0.716147	−	0.697949i	$-0.245904\pi$
$809$	40.7386	1.43229	0.716147	+	0.697949i	$0.245904\pi$
$810$	0	0
$811$	−1.93087	−0.0678020	−0.0339010	−	0.999425i	$-0.510793\pi$
$811$	−1.93087	−0.0678020	−0.0339010	+	0.999425i	$0.510793\pi$
$812$	0	0
$813$	10.2462	0.359350
$814$	0	0
$815$	20.8078	0.728864
$816$	0	0
$817$	−4.49242	−0.157170
$818$	0	0
$819$	18.2462	0.637574
$820$	0	0
$821$	−14.4924	−0.505789	−0.252895	−	0.967494i	$-0.581383\pi$
$821$	−14.4924	−0.505789	−0.252895	+	0.967494i	$0.581383\pi$
$822$	0	0
$823$	−12.4924	−0.435458	−0.217729	−	0.976009i	$-0.569865\pi$
$823$	−12.4924	−0.435458	−0.217729	+	0.976009i	$0.569865\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.8617	1.03839	0.519197	−	0.854654i	$-0.326231\pi$
$827$	29.8617	1.03839	0.519197	+	0.854654i	$0.326231\pi$
$828$	0	0
$829$	−32.8769	−1.14186	−0.570931	−	0.820998i	$-0.693418\pi$
$829$	−32.8769	−1.14186	−0.570931	+	0.820998i	$0.693418\pi$
$830$	0	0
$831$	−56.9848	−1.97678
$832$	0	0
$833$	106.108	3.67642
$834$	0	0
$835$	1.68466	0.0583000
$836$	0	0
$837$	−2.06913	−0.0715196
$838$	0	0
$839$	−19.5076	−0.673476	−0.336738	−	0.941598i	$-0.609324\pi$
$839$	−19.5076	−0.673476	−0.336738	+	0.941598i	$0.609324\pi$
$840$	0	0
$841$	44.3002	1.52759
$842$	0	0
$843$	21.1231	0.727518
$844$	0	0
$845$	11.7386	0.403821
$846$	0	0
$847$	0	0
$848$	0	0
$849$	15.3693	0.527474
$850$	0	0
$851$	−8.35416	−0.286377
$852$	0	0
$853$	37.1231	1.27107	0.635535	−	0.772072i	$-0.280779\pi$
$853$	37.1231	1.27107	0.635535	+	0.772072i	$0.280779\pi$
$854$	0	0
$855$	−5.12311	−0.175207
$856$	0	0
$857$	−12.8078	−0.437505	−0.218752	−	0.975780i	$-0.570199\pi$
$857$	−12.8078	−0.437505	−0.218752	+	0.975780i	$0.570199\pi$
$858$	0	0
$859$	−23.8617	−0.814152	−0.407076	−	0.913394i	$-0.633452\pi$
$859$	−23.8617	−0.814152	−0.407076	+	0.913394i	$0.633452\pi$
$860$	0	0
$861$	−143.093	−4.87659
$862$	0	0
$863$	−29.1231	−0.991362	−0.495681	−	0.868505i	$-0.665081\pi$
$863$	−29.1231	−0.991362	−0.495681	+	0.868505i	$0.665081\pi$
$864$	0	0
$865$	10.8769	0.369826
$866$	0	0
$867$	−107.723	−3.65848
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−17.3693	−0.587862
$874$	0	0
$875$	−4.56155	−0.154209
$876$	0	0
$877$	25.6155	0.864975	0.432487	−	0.901640i	$-0.357636\pi$
$877$	25.6155	0.864975	0.432487	+	0.901640i	$0.357636\pi$
$878$	0	0
$879$	43.8617	1.47942
$880$	0	0
$881$	5.50758	0.185555	0.0927775	−	0.995687i	$-0.470425\pi$
$881$	5.50758	0.185555	0.0927775	+	0.995687i	$0.470425\pi$
$882$	0	0
$883$	27.5464	0.927010	0.463505	−	0.886094i	$-0.346592\pi$
$883$	27.5464	0.927010	0.463505	+	0.886094i	$0.346592\pi$
$884$	0	0
$885$	−2.87689	−0.0967057
$886$	0	0
$887$	−27.1231	−0.910705	−0.455352	−	0.890311i	$-0.650487\pi$
$887$	−27.1231	−0.910705	−0.455352	+	0.890311i	$0.650487\pi$
$888$	0	0
$889$	22.2462	0.746114
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.3542	0.547271
$894$	0	0
$895$	14.2462	0.476198
$896$	0	0
$897$	3.23106	0.107882
$898$	0	0
$899$	−12.3153	−0.410740
$900$	0	0
$901$	74.4233	2.47940
$902$	0	0
$903$	36.4924	1.21439
$904$	0	0
$905$	8.24621	0.274113
$906$	0	0
$907$	43.6847	1.45053	0.725263	−	0.688472i	$-0.241718\pi$
$907$	43.6847	1.45053	0.725263	+	0.688472i	$0.241718\pi$
$908$	0	0
$909$	7.12311	0.236259
$910$	0	0
$911$	−41.4384	−1.37292	−0.686459	−	0.727169i	$-0.740836\pi$
$911$	−41.4384	−1.37292	−0.686459	+	0.727169i	$0.740836\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	32.1771	1.06374
$916$	0	0
$917$	−76.6695	−2.53185
$918$	0	0
$919$	−35.2311	−1.16217	−0.581083	−	0.813845i	$-0.697371\pi$
$919$	−35.2311	−1.16217	−0.581083	+	0.813845i	$0.697371\pi$
$920$	0	0
$921$	−46.1080	−1.51931
$922$	0	0
$923$	−4.13826	−0.136213
$924$	0	0
$925$	7.43845	0.244575
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33.5464	1.10062	0.550311	−	0.834960i	$-0.314509\pi$
$929$	33.5464	1.10062	0.550311	+	0.834960i	$0.314509\pi$
$930$	0	0
$931$	19.8617	0.650942
$932$	0	0
$933$	−24.1771	−0.791522
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.6155	1.62087	0.810434	−	0.585829i	$-0.199231\pi$
$937$	49.6155	1.62087	0.810434	+	0.585829i	$0.199231\pi$
$938$	0	0
$939$	−63.3693	−2.06798
$940$	0	0
$941$	−9.05398	−0.295151	−0.147576	−	0.989051i	$-0.547147\pi$
$941$	−9.05398	−0.295151	−0.147576	+	0.989051i	$0.547147\pi$
$942$	0	0
$943$	−13.7538	−0.447885
$944$	0	0
$945$	6.56155	0.213447
$946$	0	0
$947$	−12.9460	−0.420689	−0.210345	−	0.977627i	$-0.567459\pi$
$947$	−12.9460	−0.420689	−0.210345	+	0.977627i	$0.567459\pi$
$948$	0	0
$949$	−1.26137	−0.0409457
$950$	0	0
$951$	−16.1771	−0.524578
$952$	0	0
$953$	6.56155	0.212550	0.106275	−	0.994337i	$-0.466108\pi$
$953$	6.56155	0.212550	0.106275	+	0.994337i	$0.466108\pi$
$954$	0	0
$955$	10.2462	0.331560
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.4924	−1.04924
$960$	0	0
$961$	−28.9309	−0.933254
$962$	0	0
$963$	25.3693	0.817515
$964$	0	0
$965$	−21.4384	−0.690128
$966$	0	0
$967$	−12.5616	−0.403952	−0.201976	−	0.979390i	$-0.564736\pi$
$967$	−12.5616	−0.403952	−0.201976	+	0.979390i	$0.564736\pi$
$968$	0	0
$969$	−28.3153	−0.909620
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−18.2462	−0.584947
$974$	0	0
$975$	−2.87689	−0.0921344
$976$	0	0
$977$	−21.8617	−0.699419	−0.349710	−	0.936858i	$-0.613720\pi$
$977$	−21.8617	−0.699419	−0.349710	+	0.936858i	$0.613720\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.1231	0.482844
$982$	0	0
$983$	16.6307	0.530436	0.265218	−	0.964188i	$-0.414556\pi$
$983$	16.6307	0.530436	0.265218	+	0.964188i	$0.414556\pi$
$984$	0	0
$985$	22.7386	0.724513
$986$	0	0
$987$	−132.847	−4.22855
$988$	0	0
$989$	3.50758	0.111534
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−48.0000	−1.52323
$994$	0	0
$995$	−4.31534	−0.136806
$996$	0	0
$997$	−1.26137	−0.0399479	−0.0199739	−	0.999801i	$-0.506358\pi$
$997$	−1.26137	−0.0399479	−0.0199739	+	0.999801i	$0.506358\pi$
$998$	0	0
$999$	−10.6998	−0.338527

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bm.1.1		2
4.3	odd	2		4840.2.a.m.1.2		2
11.10	odd	2		880.2.a.k.1.1		2
33.32	even	2		7920.2.a.by.1.1		2
44.43	even	2		440.2.a.g.1.2	✓	2
55.32	even	4		4400.2.b.w.4049.4		4
55.43	even	4		4400.2.b.w.4049.1		4
55.54	odd	2		4400.2.a.bt.1.2		2
88.21	odd	2		3520.2.a.br.1.2		2
88.43	even	2		3520.2.a.bm.1.1		2
132.131	odd	2		3960.2.a.bf.1.2		2
220.43	odd	4		2200.2.b.f.1849.4		4
220.87	odd	4		2200.2.b.f.1849.1		4
220.219	even	2		2200.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.g.1.2	✓	2	44.43	even	2
880.2.a.k.1.1		2	11.10	odd	2
2200.2.a.l.1.1		2	220.219	even	2
2200.2.b.f.1849.1		4	220.87	odd	4
2200.2.b.f.1849.4		4	220.43	odd	4
3520.2.a.bm.1.1		2	88.43	even	2
3520.2.a.br.1.2		2	88.21	odd	2
3960.2.a.bf.1.2		2	132.131	odd	2
4400.2.a.bt.1.2		2	55.54	odd	2
4400.2.b.w.4049.1		4	55.43	even	4
4400.2.b.w.4049.4		4	55.32	even	4
4840.2.a.m.1.2		2	4.3	odd	2
7920.2.a.by.1.1		2	33.32	even	2
9680.2.a.bm.1.1		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +1.12311 q^{13} +2.56155 q^{15} +7.68466 q^{17} +1.43845 q^{19} -11.6847 q^{21} -1.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} -8.56155 q^{29} +1.43845 q^{31} -4.56155 q^{35} +7.43845 q^{37} -2.87689 q^{39} +12.2462 q^{41} -3.12311 q^{43} -3.56155 q^{45} +11.3693 q^{47} +13.8078 q^{49} -19.6847 q^{51} +9.68466 q^{53} -3.68466 q^{57} -1.12311 q^{59} +12.5616 q^{61} +16.2462 q^{63} -1.12311 q^{65} +2.87689 q^{69} -3.68466 q^{71} -1.12311 q^{73} -2.56155 q^{75} -11.3693 q^{79} -7.00000 q^{81} -6.00000 q^{83} -7.68466 q^{85} +21.9309 q^{87} +9.68466 q^{89} +5.12311 q^{91} -3.68466 q^{93} -1.43845 q^{95} -4.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 6 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} - 11 q^{21} + 6 q^{23} + 2 q^{25} - 7 q^{27} - 13 q^{29} + 7 q^{31} - 5 q^{35} + 19 q^{37} - 14 q^{39} + 8 q^{41} + 2 q^{43}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 5 * q^7 + 3 * q^9 - 6 * q^13 + q^15 + 3 * q^17 + 7 * q^19 - 11 * q^21 + 6 * q^23 + 2 * q^25 - 7 * q^27 - 13 * q^29 + 7 * q^31 - 5 * q^35 + 19 * q^37 - 14 * q^39 + 8 * q^41 + 2 * q^43 - 3 * q^45 - 2 * q^47 + 7 * q^49 - 27 * q^51 + 7 * q^53 + 5 * q^57 + 6 * q^59 + 21 * q^61 + 16 * q^63 + 6 * q^65 + 14 * q^69 + 5 * q^71 + 6 * q^73 - q^75 + 2 * q^79 - 14 * q^81 - 12 * q^83 - 3 * q^85 + 15 * q^87 + 7 * q^89 + 2 * q^91 + 5 * q^93 - 7 * q^95 - 18 * q^97

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