LMFDB - Embedded newform 2888.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] 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("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-2,0,4,0,-3,0,-11,0,-2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} -3.23607 q^{5} -0.236068 q^{7} -0.381966 q^{9} -4.38197 q^{11} +3.47214 q^{13} +5.23607 q^{15} +7.23607 q^{17} +0.381966 q^{21} +5.85410 q^{23} +5.47214 q^{25} +5.47214 q^{27} -2.85410 q^{29} -4.38197 q^{31} +7.09017 q^{33} +0.763932 q^{35} -6.38197 q^{37} -5.61803 q^{39} +7.94427 q^{41} -0.618034 q^{43} +1.23607 q^{45} +12.7082 q^{47} -6.94427 q^{49} -11.7082 q^{51} +3.85410 q^{53} +14.1803 q^{55} -5.38197 q^{59} +6.70820 q^{61} +0.0901699 q^{63} -11.2361 q^{65} -6.70820 q^{67} -9.47214 q^{69} -15.1803 q^{71} +7.76393 q^{73} -8.85410 q^{75} +1.03444 q^{77} -6.00000 q^{79} -7.70820 q^{81} -14.9443 q^{83} -23.4164 q^{85} +4.61803 q^{87} -6.70820 q^{89} -0.819660 q^{91} +7.09017 q^{93} -16.3262 q^{97} +1.67376 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 4 q^{7} - 3 q^{9} - 11 q^{11} - 2 q^{13} + 6 q^{15} + 10 q^{17} + 3 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + q^{29} - 11 q^{31} + 3 q^{33} + 6 q^{35} - 15 q^{37} - 9 q^{39} - 2 q^{41}+ \cdots + 19 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 4 * q^7 - 3 * q^9 - 11 * q^11 - 2 * q^13 + 6 * q^15 + 10 * q^17 + 3 * q^21 + 5 * q^23 + 2 * q^25 + 2 * q^27 + q^29 - 11 * q^31 + 3 * q^33 + 6 * q^35 - 15 * q^37 - 9 * q^39 - 2 * q^41 + q^43 - 2 * q^45 + 12 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 6 * q^55 - 13 * q^59 - 11 * q^63 - 18 * q^65 - 10 * q^69 - 8 * q^71 + 20 * q^73 - 11 * q^75 - 27 * q^77 - 12 * q^79 - 2 * q^81 - 12 * q^83 - 20 * q^85 + 7 * q^87 - 24 * q^91 + 3 * q^93 - 17 * q^97 + 19 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−4.38197	−1.32121	−0.660606	−	0.750733i	$-0.729701\pi$
$11$	−4.38197	−1.32121	−0.660606	+	0.750733i	$0.729701\pi$
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	5.23607	1.35195
$16$	0	0
$17$	7.23607	1.75500	0.877502	−	0.479573i	$-0.159208\pi$
$17$	7.23607	1.75500	0.877502	+	0.479573i	$0.159208\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.381966	0.0833518
$22$	0	0
$23$	5.85410	1.22066	0.610332	−	0.792145i	$-0.291036\pi$
$23$	5.85410	1.22066	0.610332	+	0.792145i	$0.291036\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−2.85410	−0.529993	−0.264997	−	0.964249i	$-0.585371\pi$
$29$	−2.85410	−0.529993	−0.264997	+	0.964249i	$0.585371\pi$
$30$	0	0
$31$	−4.38197	−0.787024	−0.393512	−	0.919319i	$-0.628740\pi$
$31$	−4.38197	−0.787024	−0.393512	+	0.919319i	$0.628740\pi$
$32$	0	0
$33$	7.09017	1.23424
$34$	0	0
$35$	0.763932	0.129128
$36$	0	0
$37$	−6.38197	−1.04919	−0.524594	−	0.851352i	$-0.675783\pi$
$37$	−6.38197	−1.04919	−0.524594	+	0.851352i	$0.675783\pi$
$38$	0	0
$39$	−5.61803	−0.899605
$40$	0	0
$41$	7.94427	1.24069	0.620343	−	0.784330i	$-0.286993\pi$
$41$	7.94427	1.24069	0.620343	+	0.784330i	$0.286993\pi$
$42$	0	0
$43$	−0.618034	−0.0942493	−0.0471246	−	0.998889i	$-0.515006\pi$
$43$	−0.618034	−0.0942493	−0.0471246	+	0.998889i	$0.515006\pi$
$44$	0	0
$45$	1.23607	0.184262
$46$	0	0
$47$	12.7082	1.85368	0.926841	−	0.375454i	$-0.122513\pi$
$47$	12.7082	1.85368	0.926841	+	0.375454i	$0.122513\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	−11.7082	−1.63948
$52$	0	0
$53$	3.85410	0.529402	0.264701	−	0.964331i	$-0.414727\pi$
$53$	3.85410	0.529402	0.264701	+	0.964331i	$0.414727\pi$
$54$	0	0
$55$	14.1803	1.91208
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.38197	−0.700672	−0.350336	−	0.936624i	$-0.613933\pi$
$59$	−5.38197	−0.700672	−0.350336	+	0.936624i	$0.613933\pi$
$60$	0	0
$61$	6.70820	0.858898	0.429449	−	0.903091i	$-0.358708\pi$
$61$	6.70820	0.858898	0.429449	+	0.903091i	$0.358708\pi$
$62$	0	0
$63$	0.0901699	0.0113603
$64$	0	0
$65$	−11.2361	−1.39366
$66$	0	0
$67$	−6.70820	−0.819538	−0.409769	−	0.912189i	$-0.634391\pi$
$67$	−6.70820	−0.819538	−0.409769	+	0.912189i	$0.634391\pi$
$68$	0	0
$69$	−9.47214	−1.14031
$70$	0	0
$71$	−15.1803	−1.80157	−0.900787	−	0.434260i	$-0.857010\pi$
$71$	−15.1803	−1.80157	−0.900787	+	0.434260i	$0.857010\pi$
$72$	0	0
$73$	7.76393	0.908700	0.454350	−	0.890823i	$-0.349872\pi$
$73$	7.76393	0.908700	0.454350	+	0.890823i	$0.349872\pi$
$74$	0	0
$75$	−8.85410	−1.02238
$76$	0	0
$77$	1.03444	0.117886
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−14.9443	−1.64035	−0.820173	−	0.572115i	$-0.806123\pi$
$83$	−14.9443	−1.64035	−0.820173	+	0.572115i	$0.806123\pi$
$84$	0	0
$85$	−23.4164	−2.53987
$86$	0	0
$87$	4.61803	0.495105
$88$	0	0
$89$	−6.70820	−0.711068	−0.355534	−	0.934663i	$-0.615701\pi$
$89$	−6.70820	−0.711068	−0.355534	+	0.934663i	$0.615701\pi$
$90$	0	0
$91$	−0.819660	−0.0859237
$92$	0	0
$93$	7.09017	0.735216
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.3262	−1.65768	−0.828839	−	0.559487i	$-0.810998\pi$
$97$	−16.3262	−1.65768	−0.828839	+	0.559487i	$0.810998\pi$
$98$	0	0
$99$	1.67376	0.168219
$100$	0	0
$101$	−0.236068	−0.0234896	−0.0117448	−	0.999931i	$-0.503739\pi$
$101$	−0.236068	−0.0234896	−0.0117448	+	0.999931i	$0.503739\pi$
$102$	0	0
$103$	−1.85410	−0.182690	−0.0913450	−	0.995819i	$-0.529117\pi$
$103$	−1.85410	−0.182690	−0.0913450	+	0.995819i	$0.529117\pi$
$104$	0	0
$105$	−1.23607	−0.120628
$106$	0	0
$107$	16.7082	1.61524	0.807622	−	0.589701i	$-0.200754\pi$
$107$	16.7082	1.61524	0.807622	+	0.589701i	$0.200754\pi$
$108$	0	0
$109$	0.236068	0.0226112	0.0113056	−	0.999936i	$-0.496401\pi$
$109$	0.236068	0.0226112	0.0113056	+	0.999936i	$0.496401\pi$
$110$	0	0
$111$	10.3262	0.980123
$112$	0	0
$113$	19.7082	1.85399	0.926996	−	0.375071i	$-0.122382\pi$
$113$	19.7082	1.85399	0.926996	+	0.375071i	$0.122382\pi$
$114$	0	0
$115$	−18.9443	−1.76656
$116$	0	0
$117$	−1.32624	−0.122611
$118$	0	0
$119$	−1.70820	−0.156591
$120$	0	0
$121$	8.20163	0.745602
$122$	0	0
$123$	−12.8541	−1.15902
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	−5.38197	−0.470225	−0.235112	−	0.971968i	$-0.575546\pi$
$131$	−5.38197	−0.470225	−0.235112	+	0.971968i	$0.575546\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−17.7082	−1.52408
$136$	0	0
$137$	6.52786	0.557713	0.278857	−	0.960333i	$-0.410045\pi$
$137$	6.52786	0.557713	0.278857	+	0.960333i	$0.410045\pi$
$138$	0	0
$139$	−4.14590	−0.351650	−0.175825	−	0.984421i	$-0.556259\pi$
$139$	−4.14590	−0.351650	−0.175825	+	0.984421i	$0.556259\pi$
$140$	0	0
$141$	−20.5623	−1.73166
$142$	0	0
$143$	−15.2148	−1.27232
$144$	0	0
$145$	9.23607	0.767014
$146$	0	0
$147$	11.2361	0.926735
$148$	0	0
$149$	15.6180	1.27948	0.639740	−	0.768592i	$-0.279042\pi$
$149$	15.6180	1.27948	0.639740	+	0.768592i	$0.279042\pi$
$150$	0	0
$151$	−8.90983	−0.725072	−0.362536	−	0.931970i	$-0.618089\pi$
$151$	−8.90983	−0.725072	−0.362536	+	0.931970i	$0.618089\pi$
$152$	0	0
$153$	−2.76393	−0.223451
$154$	0	0
$155$	14.1803	1.13899
$156$	0	0
$157$	−10.7984	−0.861804	−0.430902	−	0.902399i	$-0.641805\pi$
$157$	−10.7984	−0.861804	−0.430902	+	0.902399i	$0.641805\pi$
$158$	0	0
$159$	−6.23607	−0.494552
$160$	0	0
$161$	−1.38197	−0.108914
$162$	0	0
$163$	12.4164	0.972528	0.486264	−	0.873812i	$-0.338359\pi$
$163$	12.4164	0.972528	0.486264	+	0.873812i	$0.338359\pi$
$164$	0	0
$165$	−22.9443	−1.78621
$166$	0	0
$167$	21.4721	1.66156	0.830782	−	0.556598i	$-0.187894\pi$
$167$	21.4721	1.66156	0.830782	+	0.556598i	$0.187894\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.52786	0.572333	0.286166	−	0.958180i	$-0.407619\pi$
$173$	7.52786	0.572333	0.286166	+	0.958180i	$0.407619\pi$
$174$	0	0
$175$	−1.29180	−0.0976506
$176$	0	0
$177$	8.70820	0.654549
$178$	0	0
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−10.8541	−0.802358
$184$	0	0
$185$	20.6525	1.51840
$186$	0	0
$187$	−31.7082	−2.31873
$188$	0	0
$189$	−1.29180	−0.0939643
$190$	0	0
$191$	−13.9443	−1.00897	−0.504486	−	0.863420i	$-0.668318\pi$
$191$	−13.9443	−1.00897	−0.504486	+	0.863420i	$0.668318\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	18.1803	1.30192
$196$	0	0
$197$	−2.52786	−0.180103	−0.0900514	−	0.995937i	$-0.528703\pi$
$197$	−2.52786	−0.180103	−0.0900514	+	0.995937i	$0.528703\pi$
$198$	0	0
$199$	−1.05573	−0.0748386	−0.0374193	−	0.999300i	$-0.511914\pi$
$199$	−1.05573	−0.0748386	−0.0374193	+	0.999300i	$0.511914\pi$
$200$	0	0
$201$	10.8541	0.765589
$202$	0	0
$203$	0.673762	0.0472888
$204$	0	0
$205$	−25.7082	−1.79554
$206$	0	0
$207$	−2.23607	−0.155417
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−7.09017	−0.488107	−0.244054	−	0.969762i	$-0.578477\pi$
$211$	−7.09017	−0.488107	−0.244054	+	0.969762i	$0.578477\pi$
$212$	0	0
$213$	24.5623	1.68298
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	1.03444	0.0702225
$218$	0	0
$219$	−12.5623	−0.848882
$220$	0	0
$221$	25.1246	1.69006
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	−2.09017	−0.139345
$226$	0	0
$227$	−3.29180	−0.218484	−0.109242	−	0.994015i	$-0.534842\pi$
$227$	−3.29180	−0.218484	−0.109242	+	0.994015i	$0.534842\pi$
$228$	0	0
$229$	25.7984	1.70480	0.852402	−	0.522887i	$-0.175145\pi$
$229$	25.7984	1.70480	0.852402	+	0.522887i	$0.175145\pi$
$230$	0	0
$231$	−1.67376	−0.110125
$232$	0	0
$233$	12.5279	0.820728	0.410364	−	0.911922i	$-0.365402\pi$
$233$	12.5279	0.820728	0.410364	+	0.911922i	$0.365402\pi$
$234$	0	0
$235$	−41.1246	−2.68267
$236$	0	0
$237$	9.70820	0.630616
$238$	0	0
$239$	−27.2705	−1.76398	−0.881991	−	0.471266i	$-0.843797\pi$
$239$	−27.2705	−1.76398	−0.881991	+	0.471266i	$0.843797\pi$
$240$	0	0
$241$	−15.7639	−1.01544	−0.507722	−	0.861521i	$-0.669512\pi$
$241$	−15.7639	−1.01544	−0.507722	+	0.861521i	$0.669512\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	22.4721	1.43569
$246$	0	0
$247$	0	0
$248$	0	0
$249$	24.1803	1.53237
$250$	0	0
$251$	−20.2361	−1.27729	−0.638645	−	0.769502i	$-0.720505\pi$
$251$	−20.2361	−1.27729	−0.638645	+	0.769502i	$0.720505\pi$
$252$	0	0
$253$	−25.6525	−1.61276
$254$	0	0
$255$	37.8885	2.37267
$256$	0	0
$257$	7.41641	0.462623	0.231311	−	0.972880i	$-0.425698\pi$
$257$	7.41641	0.462623	0.231311	+	0.972880i	$0.425698\pi$
$258$	0	0
$259$	1.50658	0.0936142
$260$	0	0
$261$	1.09017	0.0674798
$262$	0	0
$263$	−4.47214	−0.275764	−0.137882	−	0.990449i	$-0.544029\pi$
$263$	−4.47214	−0.275764	−0.137882	+	0.990449i	$0.544029\pi$
$264$	0	0
$265$	−12.4721	−0.766157
$266$	0	0
$267$	10.8541	0.664260
$268$	0	0
$269$	−14.3262	−0.873486	−0.436743	−	0.899586i	$-0.643868\pi$
$269$	−14.3262	−0.873486	−0.436743	+	0.899586i	$0.643868\pi$
$270$	0	0
$271$	−21.6180	−1.31320	−0.656601	−	0.754238i	$-0.728006\pi$
$271$	−21.6180	−1.31320	−0.656601	+	0.754238i	$0.728006\pi$
$272$	0	0
$273$	1.32624	0.0802676
$274$	0	0
$275$	−23.9787	−1.44597
$276$	0	0
$277$	−19.4164	−1.16662	−0.583309	−	0.812250i	$-0.698242\pi$
$277$	−19.4164	−1.16662	−0.583309	+	0.812250i	$0.698242\pi$
$278$	0	0
$279$	1.67376	0.100206
$280$	0	0
$281$	−1.90983	−0.113931	−0.0569655	−	0.998376i	$-0.518142\pi$
$281$	−1.90983	−0.113931	−0.0569655	+	0.998376i	$0.518142\pi$
$282$	0	0
$283$	5.56231	0.330645	0.165322	−	0.986240i	$-0.447134\pi$
$283$	5.56231	0.330645	0.165322	+	0.986240i	$0.447134\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.87539	−0.110701
$288$	0	0
$289$	35.3607	2.08004
$290$	0	0
$291$	26.4164	1.54856
$292$	0	0
$293$	−1.85410	−0.108318	−0.0541589	−	0.998532i	$-0.517248\pi$
$293$	−1.85410	−0.108318	−0.0541589	+	0.998532i	$0.517248\pi$
$294$	0	0
$295$	17.4164	1.01402
$296$	0	0
$297$	−23.9787	−1.39139
$298$	0	0
$299$	20.3262	1.17550
$300$	0	0
$301$	0.145898	0.00840942
$302$	0	0
$303$	0.381966	0.0219434
$304$	0	0
$305$	−21.7082	−1.24301
$306$	0	0
$307$	−34.7426	−1.98287	−0.991434	−	0.130610i	$-0.958306\pi$
$307$	−34.7426	−1.98287	−0.991434	+	0.130610i	$0.958306\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	4.29180	0.243365	0.121683	−	0.992569i	$-0.461171\pi$
$311$	4.29180	0.243365	0.121683	+	0.992569i	$0.461171\pi$
$312$	0	0
$313$	17.6180	0.995830	0.497915	−	0.867226i	$-0.334099\pi$
$313$	17.6180	0.995830	0.497915	+	0.867226i	$0.334099\pi$
$314$	0	0
$315$	−0.291796	−0.0164408
$316$	0	0
$317$	6.94427	0.390029	0.195015	−	0.980800i	$-0.437525\pi$
$317$	6.94427	0.390029	0.195015	+	0.980800i	$0.437525\pi$
$318$	0	0
$319$	12.5066	0.700234
$320$	0	0
$321$	−27.0344	−1.50892
$322$	0	0
$323$	0	0
$324$	0	0
$325$	19.0000	1.05393
$326$	0	0
$327$	−0.381966	−0.0211228
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−16.4721	−0.905390	−0.452695	−	0.891665i	$-0.649537\pi$
$331$	−16.4721	−0.905390	−0.452695	+	0.891665i	$0.649537\pi$
$332$	0	0
$333$	2.43769	0.133585
$334$	0	0
$335$	21.7082	1.18605
$336$	0	0
$337$	3.81966	0.208070	0.104035	−	0.994574i	$-0.466825\pi$
$337$	3.81966	0.208070	0.104035	+	0.994574i	$0.466825\pi$
$338$	0	0
$339$	−31.8885	−1.73195
$340$	0	0
$341$	19.2016	1.03983
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	30.6525	1.65027
$346$	0	0
$347$	−6.94427	−0.372788	−0.186394	−	0.982475i	$-0.559680\pi$
$347$	−6.94427	−0.372788	−0.186394	+	0.982475i	$0.559680\pi$
$348$	0	0
$349$	21.8541	1.16982	0.584912	−	0.811097i	$-0.301129\pi$
$349$	21.8541	1.16982	0.584912	+	0.811097i	$0.301129\pi$
$350$	0	0
$351$	19.0000	1.01414
$352$	0	0
$353$	−32.2705	−1.71759	−0.858793	−	0.512323i	$-0.828785\pi$
$353$	−32.2705	−1.71759	−0.858793	+	0.512323i	$0.828785\pi$
$354$	0	0
$355$	49.1246	2.60726
$356$	0	0
$357$	2.76393	0.146283
$358$	0	0
$359$	−6.20163	−0.327309	−0.163655	−	0.986518i	$-0.552328\pi$
$359$	−6.20163	−0.327309	−0.163655	+	0.986518i	$0.552328\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−13.2705	−0.696521
$364$	0	0
$365$	−25.1246	−1.31508
$366$	0	0
$367$	−0.708204	−0.0369679	−0.0184840	−	0.999829i	$-0.505884\pi$
$367$	−0.708204	−0.0369679	−0.0184840	+	0.999829i	$0.505884\pi$
$368$	0	0
$369$	−3.03444	−0.157967
$370$	0	0
$371$	−0.909830	−0.0472360
$372$	0	0
$373$	−6.88854	−0.356675	−0.178338	−	0.983969i	$-0.557072\pi$
$373$	−6.88854	−0.356675	−0.178338	+	0.983969i	$0.557072\pi$
$374$	0	0
$375$	2.47214	0.127661
$376$	0	0
$377$	−9.90983	−0.510382
$378$	0	0
$379$	−3.12461	−0.160501	−0.0802503	−	0.996775i	$-0.525572\pi$
$379$	−3.12461	−0.160501	−0.0802503	+	0.996775i	$0.525572\pi$
$380$	0	0
$381$	14.5623	0.746050
$382$	0	0
$383$	6.96556	0.355923	0.177962	−	0.984037i	$-0.443050\pi$
$383$	6.96556	0.355923	0.177962	+	0.984037i	$0.443050\pi$
$384$	0	0
$385$	−3.34752	−0.170606
$386$	0	0
$387$	0.236068	0.0120000
$388$	0	0
$389$	16.3262	0.827773	0.413887	−	0.910328i	$-0.364171\pi$
$389$	16.3262	0.827773	0.413887	+	0.910328i	$0.364171\pi$
$390$	0	0
$391$	42.3607	2.14227
$392$	0	0
$393$	8.70820	0.439271
$394$	0	0
$395$	19.4164	0.976946
$396$	0	0
$397$	11.9443	0.599466	0.299733	−	0.954023i	$-0.403102\pi$
$397$	11.9443	0.599466	0.299733	+	0.954023i	$0.403102\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.8885	−0.593686	−0.296843	−	0.954926i	$-0.595934\pi$
$401$	−11.8885	−0.593686	−0.296843	+	0.954926i	$0.595934\pi$
$402$	0	0
$403$	−15.2148	−0.757902
$404$	0	0
$405$	24.9443	1.23949
$406$	0	0
$407$	27.9656	1.38620
$408$	0	0
$409$	−0.291796	−0.0144284	−0.00721419	−	0.999974i	$-0.502296\pi$
$409$	−0.291796	−0.0144284	−0.00721419	+	0.999974i	$0.502296\pi$
$410$	0	0
$411$	−10.5623	−0.521000
$412$	0	0
$413$	1.27051	0.0625177
$414$	0	0
$415$	48.3607	2.37393
$416$	0	0
$417$	6.70820	0.328502
$418$	0	0
$419$	−0.944272	−0.0461307	−0.0230654	−	0.999734i	$-0.507343\pi$
$419$	−0.944272	−0.0461307	−0.0230654	+	0.999734i	$0.507343\pi$
$420$	0	0
$421$	−6.05573	−0.295138	−0.147569	−	0.989052i	$-0.547145\pi$
$421$	−6.05573	−0.295138	−0.147569	+	0.989052i	$0.547145\pi$
$422$	0	0
$423$	−4.85410	−0.236015
$424$	0	0
$425$	39.5967	1.92072
$426$	0	0
$427$	−1.58359	−0.0766354
$428$	0	0
$429$	24.6180	1.18857
$430$	0	0
$431$	12.8885	0.620819	0.310410	−	0.950603i	$-0.399534\pi$
$431$	12.8885	0.620819	0.310410	+	0.950603i	$0.399534\pi$
$432$	0	0
$433$	−19.9787	−0.960116	−0.480058	−	0.877237i	$-0.659384\pi$
$433$	−19.9787	−0.960116	−0.480058	+	0.877237i	$0.659384\pi$
$434$	0	0
$435$	−14.9443	−0.716523
$436$	0	0
$437$	0	0
$438$	0	0
$439$	32.3050	1.54183	0.770916	−	0.636937i	$-0.219799\pi$
$439$	32.3050	1.54183	0.770916	+	0.636937i	$0.219799\pi$
$440$	0	0
$441$	2.65248	0.126308
$442$	0	0
$443$	−14.8197	−0.704103	−0.352052	−	0.935981i	$-0.614516\pi$
$443$	−14.8197	−0.704103	−0.352052	+	0.935981i	$0.614516\pi$
$444$	0	0
$445$	21.7082	1.02907
$446$	0	0
$447$	−25.2705	−1.19525
$448$	0	0
$449$	−26.8885	−1.26895	−0.634474	−	0.772944i	$-0.718783\pi$
$449$	−26.8885	−1.26895	−0.634474	+	0.772944i	$0.718783\pi$
$450$	0	0
$451$	−34.8115	−1.63921
$452$	0	0
$453$	14.4164	0.677342
$454$	0	0
$455$	2.65248	0.124350
$456$	0	0
$457$	11.1246	0.520387	0.260194	−	0.965556i	$-0.416214\pi$
$457$	11.1246	0.520387	0.260194	+	0.965556i	$0.416214\pi$
$458$	0	0
$459$	39.5967	1.84822
$460$	0	0
$461$	22.8328	1.06343	0.531715	−	0.846923i	$-0.321548\pi$
$461$	22.8328	1.06343	0.531715	+	0.846923i	$0.321548\pi$
$462$	0	0
$463$	29.5066	1.37129	0.685643	−	0.727938i	$-0.259521\pi$
$463$	29.5066	1.37129	0.685643	+	0.727938i	$0.259521\pi$
$464$	0	0
$465$	−22.9443	−1.06402
$466$	0	0
$467$	9.76393	0.451821	0.225910	−	0.974148i	$-0.427464\pi$
$467$	9.76393	0.451821	0.225910	+	0.974148i	$0.427464\pi$
$468$	0	0
$469$	1.58359	0.0731235
$470$	0	0
$471$	17.4721	0.805074
$472$	0	0
$473$	2.70820	0.124523
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.47214	−0.0674045
$478$	0	0
$479$	−9.38197	−0.428673	−0.214336	−	0.976760i	$-0.568759\pi$
$479$	−9.38197	−0.428673	−0.214336	+	0.976760i	$0.568759\pi$
$480$	0	0
$481$	−22.1591	−1.01037
$482$	0	0
$483$	2.23607	0.101745
$484$	0	0
$485$	52.8328	2.39901
$486$	0	0
$487$	4.76393	0.215874	0.107937	−	0.994158i	$-0.465575\pi$
$487$	4.76393	0.215874	0.107937	+	0.994158i	$0.465575\pi$
$488$	0	0
$489$	−20.0902	−0.908509
$490$	0	0
$491$	14.2705	0.644019	0.322010	−	0.946736i	$-0.395642\pi$
$491$	14.2705	0.644019	0.322010	+	0.946736i	$0.395642\pi$
$492$	0	0
$493$	−20.6525	−0.930141
$494$	0	0
$495$	−5.41641	−0.243449
$496$	0	0
$497$	3.58359	0.160746
$498$	0	0
$499$	−7.81966	−0.350056	−0.175028	−	0.984563i	$-0.556002\pi$
$499$	−7.81966	−0.350056	−0.175028	+	0.984563i	$0.556002\pi$
$500$	0	0
$501$	−34.7426	−1.55219
$502$	0	0
$503$	24.2361	1.08063	0.540316	−	0.841462i	$-0.318305\pi$
$503$	24.2361	1.08063	0.540316	+	0.841462i	$0.318305\pi$
$504$	0	0
$505$	0.763932	0.0339945
$506$	0	0
$507$	1.52786	0.0678548
$508$	0	0
$509$	−40.8673	−1.81141	−0.905705	−	0.423909i	$-0.860658\pi$
$509$	−40.8673	−1.81141	−0.905705	+	0.423909i	$0.860658\pi$
$510$	0	0
$511$	−1.83282	−0.0810790
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	−55.6869	−2.44911
$518$	0	0
$519$	−12.1803	−0.534658
$520$	0	0
$521$	−14.8541	−0.650770	−0.325385	−	0.945582i	$-0.605494\pi$
$521$	−14.8541	−0.650770	−0.325385	+	0.945582i	$0.605494\pi$
$522$	0	0
$523$	23.0689	1.00873	0.504366	−	0.863490i	$-0.331726\pi$
$523$	23.0689	1.00873	0.504366	+	0.863490i	$0.331726\pi$
$524$	0	0
$525$	2.09017	0.0912225
$526$	0	0
$527$	−31.7082	−1.38123
$528$	0	0
$529$	11.2705	0.490022
$530$	0	0
$531$	2.05573	0.0892110
$532$	0	0
$533$	27.5836	1.19478
$534$	0	0
$535$	−54.0689	−2.33760
$536$	0	0
$537$	27.5066	1.18700
$538$	0	0
$539$	30.4296	1.31069
$540$	0	0
$541$	−7.58359	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$541$	−7.58359	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$542$	0	0
$543$	19.4164	0.833238
$544$	0	0
$545$	−0.763932	−0.0327233
$546$	0	0
$547$	−12.5623	−0.537125	−0.268563	−	0.963262i	$-0.586549\pi$
$547$	−12.5623	−0.537125	−0.268563	+	0.963262i	$0.586549\pi$
$548$	0	0
$549$	−2.56231	−0.109357
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.41641	0.0602318
$554$	0	0
$555$	−33.4164	−1.41845
$556$	0	0
$557$	−0.819660	−0.0347301	−0.0173651	−	0.999849i	$-0.505528\pi$
$557$	−0.819660	−0.0347301	−0.0173651	+	0.999849i	$0.505528\pi$
$558$	0	0
$559$	−2.14590	−0.0907618
$560$	0	0
$561$	51.3050	2.16610
$562$	0	0
$563$	−37.4164	−1.57691	−0.788457	−	0.615090i	$-0.789120\pi$
$563$	−37.4164	−1.57691	−0.788457	+	0.615090i	$0.789120\pi$
$564$	0	0
$565$	−63.7771	−2.68312
$566$	0	0
$567$	1.81966	0.0764185
$568$	0	0
$569$	9.56231	0.400873	0.200436	−	0.979707i	$-0.435764\pi$
$569$	9.56231	0.400873	0.200436	+	0.979707i	$0.435764\pi$
$570$	0	0
$571$	7.32624	0.306594	0.153297	−	0.988180i	$-0.451011\pi$
$571$	7.32624	0.306594	0.153297	+	0.988180i	$0.451011\pi$
$572$	0	0
$573$	22.5623	0.942554
$574$	0	0
$575$	32.0344	1.33593
$576$	0	0
$577$	−19.7639	−0.822783	−0.411392	−	0.911459i	$-0.634957\pi$
$577$	−19.7639	−0.822783	−0.411392	+	0.911459i	$0.634957\pi$
$578$	0	0
$579$	9.70820	0.403459
$580$	0	0
$581$	3.52786	0.146360
$582$	0	0
$583$	−16.8885	−0.699452
$584$	0	0
$585$	4.29180	0.177444
$586$	0	0
$587$	−35.8328	−1.47898	−0.739489	−	0.673168i	$-0.764933\pi$
$587$	−35.8328	−1.47898	−0.739489	+	0.673168i	$0.764933\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.09017	0.168247
$592$	0	0
$593$	−14.7082	−0.603994	−0.301997	−	0.953309i	$-0.597653\pi$
$593$	−14.7082	−0.603994	−0.301997	+	0.953309i	$0.597653\pi$
$594$	0	0
$595$	5.52786	0.226620
$596$	0	0
$597$	1.70820	0.0699121
$598$	0	0
$599$	12.1246	0.495398	0.247699	−	0.968837i	$-0.420326\pi$
$599$	12.1246	0.495398	0.247699	+	0.968837i	$0.420326\pi$
$600$	0	0
$601$	−1.81966	−0.0742255	−0.0371127	−	0.999311i	$-0.511816\pi$
$601$	−1.81966	−0.0742255	−0.0371127	+	0.999311i	$0.511816\pi$
$602$	0	0
$603$	2.56231	0.104345
$604$	0	0
$605$	−26.5410	−1.07905
$606$	0	0
$607$	33.6180	1.36451	0.682257	−	0.731112i	$-0.260999\pi$
$607$	33.6180	1.36451	0.682257	+	0.731112i	$0.260999\pi$
$608$	0	0
$609$	−1.09017	−0.0441759
$610$	0	0
$611$	44.1246	1.78509
$612$	0	0
$613$	−32.3050	−1.30478	−0.652392	−	0.757881i	$-0.726235\pi$
$613$	−32.3050	−1.30478	−0.652392	+	0.757881i	$0.726235\pi$
$614$	0	0
$615$	41.5967	1.67734
$616$	0	0
$617$	25.5623	1.02910	0.514550	−	0.857460i	$-0.327959\pi$
$617$	25.5623	1.02910	0.514550	+	0.857460i	$0.327959\pi$
$618$	0	0
$619$	−15.0000	−0.602901	−0.301450	−	0.953482i	$-0.597471\pi$
$619$	−15.0000	−0.602901	−0.301450	+	0.953482i	$0.597471\pi$
$620$	0	0
$621$	32.0344	1.28550
$622$	0	0
$623$	1.58359	0.0634453
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−46.1803	−1.84133
$630$	0	0
$631$	33.5410	1.33525	0.667623	−	0.744499i	$-0.267312\pi$
$631$	33.5410	1.33525	0.667623	+	0.744499i	$0.267312\pi$
$632$	0	0
$633$	11.4721	0.455976
$634$	0	0
$635$	29.1246	1.15577
$636$	0	0
$637$	−24.1115	−0.955331
$638$	0	0
$639$	5.79837	0.229380
$640$	0	0
$641$	−2.38197	−0.0940820	−0.0470410	−	0.998893i	$-0.514979\pi$
$641$	−2.38197	−0.0940820	−0.0470410	+	0.998893i	$0.514979\pi$
$642$	0	0
$643$	−1.81966	−0.0717604	−0.0358802	−	0.999356i	$-0.511423\pi$
$643$	−1.81966	−0.0717604	−0.0358802	+	0.999356i	$0.511423\pi$
$644$	0	0
$645$	−3.23607	−0.127420
$646$	0	0
$647$	−25.0689	−0.985560	−0.492780	−	0.870154i	$-0.664019\pi$
$647$	−25.0689	−0.985560	−0.492780	+	0.870154i	$0.664019\pi$
$648$	0	0
$649$	23.5836	0.925737
$650$	0	0
$651$	−1.67376	−0.0655999
$652$	0	0
$653$	20.7984	0.813903	0.406952	−	0.913450i	$-0.366592\pi$
$653$	20.7984	0.813903	0.406952	+	0.913450i	$0.366592\pi$
$654$	0	0
$655$	17.4164	0.680515
$656$	0	0
$657$	−2.96556	−0.115697
$658$	0	0
$659$	26.3607	1.02687	0.513433	−	0.858130i	$-0.328373\pi$
$659$	26.3607	1.02687	0.513433	+	0.858130i	$0.328373\pi$
$660$	0	0
$661$	−17.4164	−0.677420	−0.338710	−	0.940891i	$-0.609991\pi$
$661$	−17.4164	−0.677420	−0.338710	+	0.940891i	$0.609991\pi$
$662$	0	0
$663$	−40.6525	−1.57881
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.7082	−0.646944
$668$	0	0
$669$	11.3262	0.437898
$670$	0	0
$671$	−29.3951	−1.13479
$672$	0	0
$673$	−0.596748	−0.0230029	−0.0115015	−	0.999934i	$-0.503661\pi$
$673$	−0.596748	−0.0230029	−0.0115015	+	0.999934i	$0.503661\pi$
$674$	0	0
$675$	29.9443	1.15256
$676$	0	0
$677$	−23.5066	−0.903431	−0.451716	−	0.892162i	$-0.649188\pi$
$677$	−23.5066	−0.903431	−0.451716	+	0.892162i	$0.649188\pi$
$678$	0	0
$679$	3.85410	0.147907
$680$	0	0
$681$	5.32624	0.204102
$682$	0	0
$683$	4.05573	0.155188	0.0775941	−	0.996985i	$-0.475276\pi$
$683$	4.05573	0.155188	0.0775941	+	0.996985i	$0.475276\pi$
$684$	0	0
$685$	−21.1246	−0.807130
$686$	0	0
$687$	−41.7426	−1.59258
$688$	0	0
$689$	13.3820	0.509812
$690$	0	0
$691$	−9.47214	−0.360337	−0.180169	−	0.983636i	$-0.557664\pi$
$691$	−9.47214	−0.360337	−0.180169	+	0.983636i	$0.557664\pi$
$692$	0	0
$693$	−0.395122	−0.0150094
$694$	0	0
$695$	13.4164	0.508913
$696$	0	0
$697$	57.4853	2.17741
$698$	0	0
$699$	−20.2705	−0.766701
$700$	0	0
$701$	−24.7984	−0.936622	−0.468311	−	0.883564i	$-0.655137\pi$
$701$	−24.7984	−0.936622	−0.468311	+	0.883564i	$0.655137\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	66.5410	2.50608
$706$	0	0
$707$	0.0557281	0.00209587
$708$	0	0
$709$	−28.3607	−1.06511	−0.532554	−	0.846396i	$-0.678768\pi$
$709$	−28.3607	−1.06511	−0.532554	+	0.846396i	$0.678768\pi$
$710$	0	0
$711$	2.29180	0.0859491
$712$	0	0
$713$	−25.6525	−0.960693
$714$	0	0
$715$	49.2361	1.84132
$716$	0	0
$717$	44.1246	1.64786
$718$	0	0
$719$	−20.6738	−0.771001	−0.385501	−	0.922708i	$-0.625971\pi$
$719$	−20.6738	−0.771001	−0.385501	+	0.922708i	$0.625971\pi$
$720$	0	0
$721$	0.437694	0.0163006
$722$	0	0
$723$	25.5066	0.948600
$724$	0	0
$725$	−15.6180	−0.580039
$726$	0	0
$727$	29.2361	1.08431	0.542153	−	0.840280i	$-0.317609\pi$
$727$	29.2361	1.08431	0.542153	+	0.840280i	$0.317609\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−4.47214	−0.165408
$732$	0	0
$733$	21.6525	0.799752	0.399876	−	0.916569i	$-0.369053\pi$
$733$	21.6525	0.799752	0.399876	+	0.916569i	$0.369053\pi$
$734$	0	0
$735$	−36.3607	−1.34118
$736$	0	0
$737$	29.3951	1.08278
$738$	0	0
$739$	49.0689	1.80503	0.902514	−	0.430660i	$-0.141719\pi$
$739$	49.0689	1.80503	0.902514	+	0.430660i	$0.141719\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.2918	1.07461	0.537306	−	0.843387i	$-0.319442\pi$
$743$	29.2918	1.07461	0.537306	+	0.843387i	$0.319442\pi$
$744$	0	0
$745$	−50.5410	−1.85168
$746$	0	0
$747$	5.70820	0.208852
$748$	0	0
$749$	−3.94427	−0.144121
$750$	0	0
$751$	11.9098	0.434596	0.217298	−	0.976105i	$-0.430276\pi$
$751$	11.9098	0.434596	0.217298	+	0.976105i	$0.430276\pi$
$752$	0	0
$753$	32.7426	1.19321
$754$	0	0
$755$	28.8328	1.04933
$756$	0	0
$757$	−22.2016	−0.806932	−0.403466	−	0.914995i	$-0.632195\pi$
$757$	−22.2016	−0.806932	−0.403466	+	0.914995i	$0.632195\pi$
$758$	0	0
$759$	41.5066	1.50659
$760$	0	0
$761$	−3.83282	−0.138939	−0.0694697	−	0.997584i	$-0.522131\pi$
$761$	−3.83282	−0.138939	−0.0694697	+	0.997584i	$0.522131\pi$
$762$	0	0
$763$	−0.0557281	−0.00201749
$764$	0	0
$765$	8.94427	0.323381
$766$	0	0
$767$	−18.6869	−0.674745
$768$	0	0
$769$	−29.6180	−1.06805	−0.534027	−	0.845468i	$-0.679322\pi$
$769$	−29.6180	−1.06805	−0.534027	+	0.845468i	$0.679322\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	23.5623	0.847477	0.423739	−	0.905785i	$-0.360718\pi$
$773$	23.5623	0.847477	0.423739	+	0.905785i	$0.360718\pi$
$774$	0	0
$775$	−23.9787	−0.861341
$776$	0	0
$777$	−2.43769	−0.0874518
$778$	0	0
$779$	0	0
$780$	0	0
$781$	66.5197	2.38026
$782$	0	0
$783$	−15.6180	−0.558143
$784$	0	0
$785$	34.9443	1.24721
$786$	0	0
$787$	19.3050	0.688147	0.344074	−	0.938943i	$-0.388193\pi$
$787$	19.3050	0.688147	0.344074	+	0.938943i	$0.388193\pi$
$788$	0	0
$789$	7.23607	0.257611
$790$	0	0
$791$	−4.65248	−0.165423
$792$	0	0
$793$	23.2918	0.827116
$794$	0	0
$795$	20.1803	0.715723
$796$	0	0
$797$	−24.0689	−0.852564	−0.426282	−	0.904590i	$-0.640177\pi$
$797$	−24.0689	−0.852564	−0.426282	+	0.904590i	$0.640177\pi$
$798$	0	0
$799$	91.9574	3.25322
$800$	0	0
$801$	2.56231	0.0905346
$802$	0	0
$803$	−34.0213	−1.20059
$804$	0	0
$805$	4.47214	0.157622
$806$	0	0
$807$	23.1803	0.815987
$808$	0	0
$809$	36.5066	1.28350	0.641751	−	0.766913i	$-0.278208\pi$
$809$	36.5066	1.28350	0.641751	+	0.766913i	$0.278208\pi$
$810$	0	0
$811$	43.7426	1.53601	0.768006	−	0.640443i	$-0.221249\pi$
$811$	43.7426	1.53601	0.768006	+	0.640443i	$0.221249\pi$
$812$	0	0
$813$	34.9787	1.22676
$814$	0	0
$815$	−40.1803	−1.40746
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.313082	0.0109400
$820$	0	0
$821$	42.7426	1.49173	0.745864	−	0.666098i	$-0.232037\pi$
$821$	42.7426	1.49173	0.745864	+	0.666098i	$0.232037\pi$
$822$	0	0
$823$	14.7639	0.514638	0.257319	−	0.966326i	$-0.417161\pi$
$823$	14.7639	0.514638	0.257319	+	0.966326i	$0.417161\pi$
$824$	0	0
$825$	38.7984	1.35079
$826$	0	0
$827$	−9.58359	−0.333254	−0.166627	−	0.986020i	$-0.553288\pi$
$827$	−9.58359	−0.333254	−0.166627	+	0.986020i	$0.553288\pi$
$828$	0	0
$829$	13.4377	0.466710	0.233355	−	0.972392i	$-0.425030\pi$
$829$	13.4377	0.466710	0.233355	+	0.972392i	$0.425030\pi$
$830$	0	0
$831$	31.4164	1.08982
$832$	0	0
$833$	−50.2492	−1.74103
$834$	0	0
$835$	−69.4853	−2.40464
$836$	0	0
$837$	−23.9787	−0.828826
$838$	0	0
$839$	−49.9230	−1.72353	−0.861766	−	0.507305i	$-0.830642\pi$
$839$	−49.9230	−1.72353	−0.861766	+	0.507305i	$0.830642\pi$
$840$	0	0
$841$	−20.8541	−0.719107
$842$	0	0
$843$	3.09017	0.106431
$844$	0	0
$845$	3.05573	0.105120
$846$	0	0
$847$	−1.93614	−0.0665266
$848$	0	0
$849$	−9.00000	−0.308879
$850$	0	0
$851$	−37.3607	−1.28071
$852$	0	0
$853$	−11.1803	−0.382808	−0.191404	−	0.981511i	$-0.561304\pi$
$853$	−11.1803	−0.382808	−0.191404	+	0.981511i	$0.561304\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.0476	−1.60712	−0.803558	−	0.595227i	$-0.797062\pi$
$857$	−47.0476	−1.60712	−0.803558	+	0.595227i	$0.797062\pi$
$858$	0	0
$859$	−36.5410	−1.24676	−0.623382	−	0.781918i	$-0.714242\pi$
$859$	−36.5410	−1.24676	−0.623382	+	0.781918i	$0.714242\pi$
$860$	0	0
$861$	3.03444	0.103414
$862$	0	0
$863$	−35.7771	−1.21787	−0.608933	−	0.793222i	$-0.708402\pi$
$863$	−35.7771	−1.21787	−0.608933	+	0.793222i	$0.708402\pi$
$864$	0	0
$865$	−24.3607	−0.828288
$866$	0	0
$867$	−57.2148	−1.94312
$868$	0	0
$869$	26.2918	0.891888
$870$	0	0
$871$	−23.2918	−0.789212
$872$	0	0
$873$	6.23607	0.211059
$874$	0	0
$875$	0.360680	0.0121932
$876$	0	0
$877$	−38.0689	−1.28549	−0.642747	−	0.766078i	$-0.722206\pi$
$877$	−38.0689	−1.28549	−0.642747	+	0.766078i	$0.722206\pi$
$878$	0	0
$879$	3.00000	0.101187
$880$	0	0
$881$	−46.7426	−1.57480	−0.787400	−	0.616443i	$-0.788573\pi$
$881$	−46.7426	−1.57480	−0.787400	+	0.616443i	$0.788573\pi$
$882$	0	0
$883$	−43.7984	−1.47393	−0.736966	−	0.675929i	$-0.763742\pi$
$883$	−43.7984	−1.47393	−0.736966	+	0.675929i	$0.763742\pi$
$884$	0	0
$885$	−28.1803	−0.947272
$886$	0	0
$887$	−19.7082	−0.661737	−0.330868	−	0.943677i	$-0.607342\pi$
$887$	−19.7082	−0.661737	−0.330868	+	0.943677i	$0.607342\pi$
$888$	0	0
$889$	2.12461	0.0712572
$890$	0	0
$891$	33.7771	1.13158
$892$	0	0
$893$	0	0
$894$	0	0
$895$	55.0132	1.83889
$896$	0	0
$897$	−32.8885	−1.09812
$898$	0	0
$899$	12.5066	0.417118
$900$	0	0
$901$	27.8885	0.929102
$902$	0	0
$903$	−0.236068	−0.00785585
$904$	0	0
$905$	38.8328	1.29085
$906$	0	0
$907$	−47.4164	−1.57444	−0.787218	−	0.616675i	$-0.788479\pi$
$907$	−47.4164	−1.57444	−0.787218	+	0.616675i	$0.788479\pi$
$908$	0	0
$909$	0.0901699	0.00299075
$910$	0	0
$911$	26.3951	0.874509	0.437255	−	0.899338i	$-0.355951\pi$
$911$	26.3951	0.874509	0.437255	+	0.899338i	$0.355951\pi$
$912$	0	0
$913$	65.4853	2.16725
$914$	0	0
$915$	35.1246	1.16118
$916$	0	0
$917$	1.27051	0.0419559
$918$	0	0
$919$	−41.2492	−1.36069	−0.680343	−	0.732894i	$-0.738169\pi$
$919$	−41.2492	−1.36069	−0.680343	+	0.732894i	$0.738169\pi$
$920$	0	0
$921$	56.2148	1.85234
$922$	0	0
$923$	−52.7082	−1.73491
$924$	0	0
$925$	−34.9230	−1.14826
$926$	0	0
$927$	0.708204	0.0232605
$928$	0	0
$929$	−37.5066	−1.23055	−0.615275	−	0.788312i	$-0.710955\pi$
$929$	−37.5066	−1.23055	−0.615275	+	0.788312i	$0.710955\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.94427	−0.227345
$934$	0	0
$935$	102.610	3.35570
$936$	0	0
$937$	−44.1591	−1.44261	−0.721307	−	0.692616i	$-0.756458\pi$
$937$	−44.1591	−1.44261	−0.721307	+	0.692616i	$0.756458\pi$
$938$	0	0
$939$	−28.5066	−0.930277
$940$	0	0
$941$	−23.9230	−0.779867	−0.389934	−	0.920843i	$-0.627502\pi$
$941$	−23.9230	−0.779867	−0.389934	+	0.920843i	$0.627502\pi$
$942$	0	0
$943$	46.5066	1.51446
$944$	0	0
$945$	4.18034	0.135986
$946$	0	0
$947$	−33.0132	−1.07278	−0.536392	−	0.843969i	$-0.680213\pi$
$947$	−33.0132	−1.07278	−0.536392	+	0.843969i	$0.680213\pi$
$948$	0	0
$949$	26.9574	0.875075
$950$	0	0
$951$	−11.2361	−0.364354
$952$	0	0
$953$	−40.7082	−1.31867	−0.659334	−	0.751850i	$-0.729162\pi$
$953$	−40.7082	−1.31867	−0.659334	+	0.751850i	$0.729162\pi$
$954$	0	0
$955$	45.1246	1.46020
$956$	0	0
$957$	−20.2361	−0.654139
$958$	0	0
$959$	−1.54102	−0.0497621
$960$	0	0
$961$	−11.7984	−0.380593
$962$	0	0
$963$	−6.38197	−0.205656
$964$	0	0
$965$	19.4164	0.625036
$966$	0	0
$967$	−8.65248	−0.278245	−0.139122	−	0.990275i	$-0.544428\pi$
$967$	−8.65248	−0.278245	−0.139122	+	0.990275i	$0.544428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	52.5623	1.68680	0.843402	−	0.537283i	$-0.180549\pi$
$971$	52.5623	1.68680	0.843402	+	0.537283i	$0.180549\pi$
$972$	0	0
$973$	0.978714	0.0313761
$974$	0	0
$975$	−30.7426	−0.984553
$976$	0	0
$977$	36.4164	1.16506	0.582532	−	0.812808i	$-0.302062\pi$
$977$	36.4164	1.16506	0.582532	+	0.812808i	$0.302062\pi$
$978$	0	0
$979$	29.3951	0.939472
$980$	0	0
$981$	−0.0901699	−0.00287890
$982$	0	0
$983$	10.3262	0.329356	0.164678	−	0.986347i	$-0.447342\pi$
$983$	10.3262	0.329356	0.164678	+	0.986347i	$0.447342\pi$
$984$	0	0
$985$	8.18034	0.260647
$986$	0	0
$987$	4.85410	0.154508
$988$	0	0
$989$	−3.61803	−0.115047
$990$	0	0
$991$	44.7984	1.42307	0.711534	−	0.702652i	$-0.248001\pi$
$991$	44.7984	1.42307	0.711534	+	0.702652i	$0.248001\pi$
$992$	0	0
$993$	26.6525	0.845791
$994$	0	0
$995$	3.41641	0.108307
$996$	0	0
$997$	−23.7082	−0.750846	−0.375423	−	0.926854i	$-0.622503\pi$
$997$	−23.7082	−0.750846	−0.375423	+	0.926854i	$0.622503\pi$
$998$	0	0
$999$	−34.9230	−1.10491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.i.1.1	✓	2
4.3	odd	2		5776.2.a.bd.1.2		2
19.18	odd	2		2888.2.a.k.1.2	yes	2
76.75	even	2		5776.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.i.1.1	✓	2	1.1	even	1	trivial
2888.2.a.k.1.2	yes	2	19.18	odd	2
5776.2.a.x.1.1		2	76.75	even	2
5776.2.a.bd.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -3.23607 q^{5} -0.236068 q^{7} -0.381966 q^{9} -4.38197 q^{11} +3.47214 q^{13} +5.23607 q^{15} +7.23607 q^{17} +0.381966 q^{21} +5.85410 q^{23} +5.47214 q^{25} +5.47214 q^{27} -2.85410 q^{29} -4.38197 q^{31} +7.09017 q^{33} +0.763932 q^{35} -6.38197 q^{37} -5.61803 q^{39} +7.94427 q^{41} -0.618034 q^{43} +1.23607 q^{45} +12.7082 q^{47} -6.94427 q^{49} -11.7082 q^{51} +3.85410 q^{53} +14.1803 q^{55} -5.38197 q^{59} +6.70820 q^{61} +0.0901699 q^{63} -11.2361 q^{65} -6.70820 q^{67} -9.47214 q^{69} -15.1803 q^{71} +7.76393 q^{73} -8.85410 q^{75} +1.03444 q^{77} -6.00000 q^{79} -7.70820 q^{81} -14.9443 q^{83} -23.4164 q^{85} +4.61803 q^{87} -6.70820 q^{89} -0.819660 q^{91} +7.09017 q^{93} -16.3262 q^{97} +1.67376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 4 q^{7} - 3 q^{9} - 11 q^{11} - 2 q^{13} + 6 q^{15} + 10 q^{17} + 3 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + q^{29} - 11 q^{31} + 3 q^{33} + 6 q^{35} - 15 q^{37} - 9 q^{39} - 2 q^{41}+ \cdots + 19 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 4 * q^7 - 3 * q^9 - 11 * q^11 - 2 * q^13 + 6 * q^15 + 10 * q^17 + 3 * q^21 + 5 * q^23 + 2 * q^25 + 2 * q^27 + q^29 - 11 * q^31 + 3 * q^33 + 6 * q^35 - 15 * q^37 - 9 * q^39 - 2 * q^41 + q^43 - 2 * q^45 + 12 * q^47 + 4 * q^49 - 10 * q^51 + q^53 + 6 * q^55 - 13 * q^59 - 11 * q^63 - 18 * q^65 - 10 * q^69 - 8 * q^71 + 20 * q^73 - 11 * q^75 - 27 * q^77 - 12 * q^79 - 2 * q^81 - 12 * q^83 - 20 * q^85 + 7 * q^87 - 24 * q^91 + 3 * q^93 - 17 * q^97 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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