LMFDB - Embedded newform 2888.2.a.i.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{3} +1.23607 q^{5} +4.23607 q^{7} -2.61803 q^{9} -6.61803 q^{11} -5.47214 q^{13} +0.763932 q^{15} +2.76393 q^{17} +2.61803 q^{21} -0.854102 q^{23} -3.47214 q^{25} -3.47214 q^{27} +3.85410 q^{29} -6.61803 q^{31} -4.09017 q^{33} +5.23607 q^{35} -8.61803 q^{37} -3.38197 q^{39} -9.94427 q^{41} +1.61803 q^{43} -3.23607 q^{45} -0.708204 q^{47} +10.9443 q^{49} +1.70820 q^{51} -2.85410 q^{53} -8.18034 q^{55} -7.61803 q^{59} -6.70820 q^{61} -11.0902 q^{63} -6.76393 q^{65} +6.70820 q^{67} -0.527864 q^{69} +7.18034 q^{71} +12.2361 q^{73} -2.14590 q^{75} -28.0344 q^{77} -6.00000 q^{79} +5.70820 q^{81} +2.94427 q^{83} +3.41641 q^{85} +2.38197 q^{87} +6.70820 q^{89} -23.1803 q^{91} -4.09017 q^{93} -0.673762 q^{97} +17.3262 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−6.61803	−1.99541	−0.997706	−	0.0676935i	$-0.978436\pi$
$11$	−6.61803	−1.99541	−0.997706	+	0.0676935i	$0.978436\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	0.763932	0.197246
$16$	0	0
$17$	2.76393	0.670352	0.335176	−	0.942156i	$-0.391204\pi$
$17$	2.76393	0.670352	0.335176	+	0.942156i	$0.391204\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	2.61803	0.571302
$22$	0	0
$23$	−0.854102	−0.178093	−0.0890463	−	0.996027i	$-0.528382\pi$
$23$	−0.854102	−0.178093	−0.0890463	+	0.996027i	$0.528382\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	3.85410	0.715689	0.357844	−	0.933781i	$-0.383512\pi$
$29$	3.85410	0.715689	0.357844	+	0.933781i	$0.383512\pi$
$30$	0	0
$31$	−6.61803	−1.18863	−0.594317	−	0.804231i	$-0.702578\pi$
$31$	−6.61803	−1.18863	−0.594317	+	0.804231i	$0.702578\pi$
$32$	0	0
$33$	−4.09017	−0.712007
$34$	0	0
$35$	5.23607	0.885057
$36$	0	0
$37$	−8.61803	−1.41680	−0.708398	−	0.705813i	$-0.750582\pi$
$37$	−8.61803	−1.41680	−0.708398	+	0.705813i	$0.750582\pi$
$38$	0	0
$39$	−3.38197	−0.541548
$40$	0	0
$41$	−9.94427	−1.55303	−0.776517	−	0.630096i	$-0.783015\pi$
$41$	−9.94427	−1.55303	−0.776517	+	0.630096i	$0.783015\pi$
$42$	0	0
$43$	1.61803	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$43$	1.61803	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$44$	0	0
$45$	−3.23607	−0.482405
$46$	0	0
$47$	−0.708204	−0.103302	−0.0516511	−	0.998665i	$-0.516448\pi$
$47$	−0.708204	−0.103302	−0.0516511	+	0.998665i	$0.516448\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	1.70820	0.239196
$52$	0	0
$53$	−2.85410	−0.392041	−0.196021	−	0.980600i	$-0.562802\pi$
$53$	−2.85410	−0.392041	−0.196021	+	0.980600i	$0.562802\pi$
$54$	0	0
$55$	−8.18034	−1.10304
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.61803	−0.991784	−0.495892	−	0.868384i	$-0.665159\pi$
$59$	−7.61803	−0.991784	−0.495892	+	0.868384i	$0.665159\pi$
$60$	0	0
$61$	−6.70820	−0.858898	−0.429449	−	0.903091i	$-0.641292\pi$
$61$	−6.70820	−0.858898	−0.429449	+	0.903091i	$0.641292\pi$
$62$	0	0
$63$	−11.0902	−1.39723
$64$	0	0
$65$	−6.76393	−0.838963
$66$	0	0
$67$	6.70820	0.819538	0.409769	−	0.912189i	$-0.365609\pi$
$67$	6.70820	0.819538	0.409769	+	0.912189i	$0.365609\pi$
$68$	0	0
$69$	−0.527864	−0.0635474
$70$	0	0
$71$	7.18034	0.852150	0.426075	−	0.904688i	$-0.359896\pi$
$71$	7.18034	0.852150	0.426075	+	0.904688i	$0.359896\pi$
$72$	0	0
$73$	12.2361	1.43212	0.716062	−	0.698037i	$-0.245943\pi$
$73$	12.2361	1.43212	0.716062	+	0.698037i	$0.245943\pi$
$74$	0	0
$75$	−2.14590	−0.247787
$76$	0	0
$77$	−28.0344	−3.19482
$78$	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
$81$	5.70820	0.634245
$82$	0	0
$83$	2.94427	0.323176	0.161588	−	0.986858i	$-0.448338\pi$
$83$	2.94427	0.323176	0.161588	+	0.986858i	$0.448338\pi$
$84$	0	0
$85$	3.41641	0.370561
$86$	0	0
$87$	2.38197	0.255374
$88$	0	0
$89$	6.70820	0.711068	0.355534	−	0.934663i	$-0.384299\pi$
$89$	6.70820	0.711068	0.355534	+	0.934663i	$0.384299\pi$
$90$	0	0
$91$	−23.1803	−2.42996
$92$	0	0
$93$	−4.09017	−0.424131
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.673762	−0.0684102	−0.0342051	−	0.999415i	$-0.510890\pi$
$97$	−0.673762	−0.0684102	−0.0342051	+	0.999415i	$0.510890\pi$
$98$	0	0
$99$	17.3262	1.74135
$100$	0	0
$101$	4.23607	0.421505	0.210752	−	0.977540i	$-0.432409\pi$
$101$	4.23607	0.421505	0.210752	+	0.977540i	$0.432409\pi$
$102$	0	0
$103$	4.85410	0.478289	0.239144	−	0.970984i	$-0.423133\pi$
$103$	4.85410	0.478289	0.239144	+	0.970984i	$0.423133\pi$
$104$	0	0
$105$	3.23607	0.315808
$106$	0	0
$107$	3.29180	0.318230	0.159115	−	0.987260i	$-0.449136\pi$
$107$	3.29180	0.318230	0.159115	+	0.987260i	$0.449136\pi$
$108$	0	0
$109$	−4.23607	−0.405742	−0.202871	−	0.979206i	$-0.565027\pi$
$109$	−4.23607	−0.405742	−0.202871	+	0.979206i	$0.565027\pi$
$110$	0	0
$111$	−5.32624	−0.505544
$112$	0	0
$113$	6.29180	0.591882	0.295941	−	0.955206i	$-0.404367\pi$
$113$	6.29180	0.591882	0.295941	+	0.955206i	$0.404367\pi$
$114$	0	0
$115$	−1.05573	−0.0984472
$116$	0	0
$117$	14.3262	1.32446
$118$	0	0
$119$	11.7082	1.07329
$120$	0	0
$121$	32.7984	2.98167
$122$	0	0
$123$	−6.14590	−0.554157
$124$	0	0
$125$	−10.4721	−0.936656
$126$	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
$131$	−7.61803	−0.665591	−0.332795	−	0.942999i	$-0.607992\pi$
$131$	−7.61803	−0.665591	−0.332795	+	0.942999i	$0.607992\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.29180	−0.369379
$136$	0	0
$137$	15.4721	1.32187	0.660937	−	0.750441i	$-0.270159\pi$
$137$	15.4721	1.32187	0.660937	+	0.750441i	$0.270159\pi$
$138$	0	0
$139$	−10.8541	−0.920633	−0.460316	−	0.887755i	$-0.652264\pi$
$139$	−10.8541	−0.920633	−0.460316	+	0.887755i	$0.652264\pi$
$140$	0	0
$141$	−0.437694	−0.0368605
$142$	0	0
$143$	36.2148	3.02843
$144$	0	0
$145$	4.76393	0.395623
$146$	0	0
$147$	6.76393	0.557880
$148$	0	0
$149$	13.3820	1.09629	0.548147	−	0.836382i	$-0.315334\pi$
$149$	13.3820	1.09629	0.548147	+	0.836382i	$0.315334\pi$
$150$	0	0
$151$	−20.0902	−1.63491	−0.817457	−	0.575989i	$-0.804617\pi$
$151$	−20.0902	−1.63491	−0.817457	+	0.575989i	$0.804617\pi$
$152$	0	0
$153$	−7.23607	−0.585001
$154$	0	0
$155$	−8.18034	−0.657061
$156$	0	0
$157$	13.7984	1.10123	0.550615	−	0.834759i	$-0.314393\pi$
$157$	13.7984	1.10123	0.550615	+	0.834759i	$0.314393\pi$
$158$	0	0
$159$	−1.76393	−0.139889
$160$	0	0
$161$	−3.61803	−0.285141
$162$	0	0
$163$	−14.4164	−1.12918	−0.564590	−	0.825371i	$-0.690966\pi$
$163$	−14.4164	−1.12918	−0.564590	+	0.825371i	$0.690966\pi$
$164$	0	0
$165$	−5.05573	−0.393588
$166$	0	0
$167$	12.5279	0.969435	0.484718	−	0.874671i	$-0.338922\pi$
$167$	12.5279	0.969435	0.484718	+	0.874671i	$0.338922\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.4721	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$173$	16.4721	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$174$	0	0
$175$	−14.7082	−1.11184
$176$	0	0
$177$	−4.70820	−0.353890
$178$	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
$183$	−4.14590	−0.306474
$184$	0	0
$185$	−10.6525	−0.783186
$186$	0	0
$187$	−18.2918	−1.33763
$188$	0	0
$189$	−14.7082	−1.06986
$190$	0	0
$191$	3.94427	0.285397	0.142699	−	0.989766i	$-0.454422\pi$
$191$	3.94427	0.285397	0.142699	+	0.989766i	$0.454422\pi$
$192$	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
$195$	−4.18034	−0.299360
$196$	0	0
$197$	−11.4721	−0.817356	−0.408678	−	0.912679i	$-0.634010\pi$
$197$	−11.4721	−0.817356	−0.408678	+	0.912679i	$0.634010\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	0	0
$201$	4.14590	0.292429
$202$	0	0
$203$	16.3262	1.14588
$204$	0	0
$205$	−12.2918	−0.858496
$206$	0	0
$207$	2.23607	0.155417
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.09017	0.281579	0.140789	−	0.990040i	$-0.455036\pi$
$211$	4.09017	0.281579	0.140789	+	0.990040i	$0.455036\pi$
$212$	0	0
$213$	4.43769	0.304066
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−28.0344	−1.90310
$218$	0	0
$219$	7.56231	0.511013
$220$	0	0
$221$	−15.1246	−1.01739
$222$	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
$225$	9.09017	0.606011
$226$	0	0
$227$	−16.7082	−1.10896	−0.554481	−	0.832196i	$-0.687083\pi$
$227$	−16.7082	−1.10896	−0.554481	+	0.832196i	$0.687083\pi$
$228$	0	0
$229$	1.20163	0.0794057	0.0397028	−	0.999212i	$-0.487359\pi$
$229$	1.20163	0.0794057	0.0397028	+	0.999212i	$0.487359\pi$
$230$	0	0
$231$	−17.3262	−1.13998
$232$	0	0
$233$	21.4721	1.40669	0.703343	−	0.710850i	$-0.251690\pi$
$233$	21.4721	1.40669	0.703343	+	0.710850i	$0.251690\pi$
$234$	0	0
$235$	−0.875388	−0.0571040
$236$	0	0
$237$	−3.70820	−0.240874
$238$	0	0
$239$	6.27051	0.405606	0.202803	−	0.979220i	$-0.434995\pi$
$239$	6.27051	0.405606	0.202803	+	0.979220i	$0.434995\pi$
$240$	0	0
$241$	−20.2361	−1.30352	−0.651760	−	0.758425i	$-0.725969\pi$
$241$	−20.2361	−1.30352	−0.651760	+	0.758425i	$0.725969\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	13.5279	0.864264
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.81966	0.115316
$250$	0	0
$251$	−15.7639	−0.995011	−0.497505	−	0.867461i	$-0.665750\pi$
$251$	−15.7639	−0.995011	−0.497505	+	0.867461i	$0.665750\pi$
$252$	0	0
$253$	5.65248	0.355368
$254$	0	0
$255$	2.11146	0.132225
$256$	0	0
$257$	−19.4164	−1.21116	−0.605581	−	0.795784i	$-0.707059\pi$
$257$	−19.4164	−1.21116	−0.605581	+	0.795784i	$0.707059\pi$
$258$	0	0
$259$	−36.5066	−2.26841
$260$	0	0
$261$	−10.0902	−0.624566
$262$	0	0
$263$	4.47214	0.275764	0.137882	−	0.990449i	$-0.455971\pi$
$263$	4.47214	0.275764	0.137882	+	0.990449i	$0.455971\pi$
$264$	0	0
$265$	−3.52786	−0.216715
$266$	0	0
$267$	4.14590	0.253725
$268$	0	0
$269$	1.32624	0.0808622	0.0404311	−	0.999182i	$-0.487127\pi$
$269$	1.32624	0.0808622	0.0404311	+	0.999182i	$0.487127\pi$
$270$	0	0
$271$	−19.3820	−1.17737	−0.588685	−	0.808362i	$-0.700354\pi$
$271$	−19.3820	−1.17737	−0.588685	+	0.808362i	$0.700354\pi$
$272$	0	0
$273$	−14.3262	−0.867063
$274$	0	0
$275$	22.9787	1.38567
$276$	0	0
$277$	7.41641	0.445609	0.222804	−	0.974863i	$-0.428479\pi$
$277$	7.41641	0.445609	0.222804	+	0.974863i	$0.428479\pi$
$278$	0	0
$279$	17.3262	1.03729
$280$	0	0
$281$	−13.0902	−0.780894	−0.390447	−	0.920625i	$-0.627680\pi$
$281$	−13.0902	−0.780894	−0.390447	+	0.920625i	$0.627680\pi$
$282$	0	0
$283$	−14.5623	−0.865639	−0.432820	−	0.901481i	$-0.642481\pi$
$283$	−14.5623	−0.865639	−0.432820	+	0.901481i	$0.642481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−42.1246	−2.48654
$288$	0	0
$289$	−9.36068	−0.550628
$290$	0	0
$291$	−0.416408	−0.0244103
$292$	0	0
$293$	4.85410	0.283580	0.141790	−	0.989897i	$-0.454714\pi$
$293$	4.85410	0.283580	0.141790	+	0.989897i	$0.454714\pi$
$294$	0	0
$295$	−9.41641	−0.548244
$296$	0	0
$297$	22.9787	1.33336
$298$	0	0
$299$	4.67376	0.270291
$300$	0	0
$301$	6.85410	0.395064
$302$	0	0
$303$	2.61803	0.150402
$304$	0	0
$305$	−8.29180	−0.474787
$306$	0	0
$307$	7.74265	0.441896	0.220948	−	0.975286i	$-0.429085\pi$
$307$	7.74265	0.441896	0.220948	+	0.975286i	$0.429085\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	0	0
$311$	17.7082	1.00414	0.502070	−	0.864827i	$-0.332572\pi$
$311$	17.7082	1.00414	0.502070	+	0.864827i	$0.332572\pi$
$312$	0	0
$313$	15.3820	0.869440	0.434720	−	0.900566i	$-0.356847\pi$
$313$	15.3820	0.869440	0.434720	+	0.900566i	$0.356847\pi$
$314$	0	0
$315$	−13.7082	−0.772370
$316$	0	0
$317$	−10.9443	−0.614692	−0.307346	−	0.951598i	$-0.599441\pi$
$317$	−10.9443	−0.614692	−0.307346	+	0.951598i	$0.599441\pi$
$318$	0	0
$319$	−25.5066	−1.42809
$320$	0	0
$321$	2.03444	0.113551
$322$	0	0
$323$	0	0
$324$	0	0
$325$	19.0000	1.05393
$326$	0	0
$327$	−2.61803	−0.144778
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−7.52786	−0.413769	−0.206884	−	0.978365i	$-0.566332\pi$
$331$	−7.52786	−0.413769	−0.206884	+	0.978365i	$0.566332\pi$
$332$	0	0
$333$	22.5623	1.23641
$334$	0	0
$335$	8.29180	0.453029
$336$	0	0
$337$	26.1803	1.42613	0.713067	−	0.701096i	$-0.247306\pi$
$337$	26.1803	1.42613	0.713067	+	0.701096i	$0.247306\pi$
$338$	0	0
$339$	3.88854	0.211197
$340$	0	0
$341$	43.7984	2.37181
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	−0.652476	−0.0351281
$346$	0	0
$347$	10.9443	0.587519	0.293760	−	0.955879i	$-0.405093\pi$
$347$	10.9443	0.587519	0.293760	+	0.955879i	$0.405093\pi$
$348$	0	0
$349$	15.1459	0.810741	0.405371	−	0.914152i	$-0.367142\pi$
$349$	15.1459	0.810741	0.405371	+	0.914152i	$0.367142\pi$
$350$	0	0
$351$	19.0000	1.01414
$352$	0	0
$353$	1.27051	0.0676224	0.0338112	−	0.999428i	$-0.489236\pi$
$353$	1.27051	0.0676224	0.0338112	+	0.999428i	$0.489236\pi$
$354$	0	0
$355$	8.87539	0.471057
$356$	0	0
$357$	7.23607	0.382973
$358$	0	0
$359$	−30.7984	−1.62548	−0.812738	−	0.582629i	$-0.802024\pi$
$359$	−30.7984	−1.62548	−0.812738	+	0.582629i	$0.802024\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	20.2705	1.06393
$364$	0	0
$365$	15.1246	0.791658
$366$	0	0
$367$	12.7082	0.663363	0.331681	−	0.943391i	$-0.392384\pi$
$367$	12.7082	0.663363	0.331681	+	0.943391i	$0.392384\pi$
$368$	0	0
$369$	26.0344	1.35530
$370$	0	0
$371$	−12.0902	−0.627690
$372$	0	0
$373$	28.8885	1.49579	0.747896	−	0.663816i	$-0.231064\pi$
$373$	28.8885	1.49579	0.747896	+	0.663816i	$0.231064\pi$
$374$	0	0
$375$	−6.47214	−0.334220
$376$	0	0
$377$	−21.0902	−1.08620
$378$	0	0
$379$	37.1246	1.90696	0.953482	−	0.301451i	$-0.0974710\pi$
$379$	37.1246	1.90696	0.953482	+	0.301451i	$0.0974710\pi$
$380$	0	0
$381$	−5.56231	−0.284966
$382$	0	0
$383$	36.0344	1.84127	0.920637	−	0.390420i	$-0.127670\pi$
$383$	36.0344	1.84127	0.920637	+	0.390420i	$0.127670\pi$
$384$	0	0
$385$	−34.6525	−1.76605
$386$	0	0
$387$	−4.23607	−0.215331
$388$	0	0
$389$	0.673762	0.0341611	0.0170805	−	0.999854i	$-0.494563\pi$
$389$	0.673762	0.0341611	0.0170805	+	0.999854i	$0.494563\pi$
$390$	0	0
$391$	−2.36068	−0.119385
$392$	0	0
$393$	−4.70820	−0.237497
$394$	0	0
$395$	−7.41641	−0.373160
$396$	0	0
$397$	−5.94427	−0.298334	−0.149167	−	0.988812i	$-0.547659\pi$
$397$	−5.94427	−0.298334	−0.149167	+	0.988812i	$0.547659\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.8885	1.19294	0.596468	−	0.802637i	$-0.296570\pi$
$401$	23.8885	1.19294	0.596468	+	0.802637i	$0.296570\pi$
$402$	0	0
$403$	36.2148	1.80399
$404$	0	0
$405$	7.05573	0.350602
$406$	0	0
$407$	57.0344	2.82709
$408$	0	0
$409$	−13.7082	−0.677827	−0.338914	−	0.940818i	$-0.610059\pi$
$409$	−13.7082	−0.677827	−0.338914	+	0.940818i	$0.610059\pi$
$410$	0	0
$411$	9.56231	0.471674
$412$	0	0
$413$	−32.2705	−1.58793
$414$	0	0
$415$	3.63932	0.178647
$416$	0	0
$417$	−6.70820	−0.328502
$418$	0	0
$419$	16.9443	0.827782	0.413891	−	0.910326i	$-0.364169\pi$
$419$	16.9443	0.827782	0.413891	+	0.910326i	$0.364169\pi$
$420$	0	0
$421$	−23.9443	−1.16697	−0.583486	−	0.812123i	$-0.698312\pi$
$421$	−23.9443	−1.16697	−0.583486	+	0.812123i	$0.698312\pi$
$422$	0	0
$423$	1.85410	0.0901495
$424$	0	0
$425$	−9.59675	−0.465511
$426$	0	0
$427$	−28.4164	−1.37517
$428$	0	0
$429$	22.3820	1.08061
$430$	0	0
$431$	−22.8885	−1.10250	−0.551251	−	0.834339i	$-0.685849\pi$
$431$	−22.8885	−1.10250	−0.551251	+	0.834339i	$0.685849\pi$
$432$	0	0
$433$	26.9787	1.29651	0.648257	−	0.761422i	$-0.275498\pi$
$433$	26.9787	1.29651	0.648257	+	0.761422i	$0.275498\pi$
$434$	0	0
$435$	2.94427	0.141167
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−30.3050	−1.44638	−0.723188	−	0.690651i	$-0.757324\pi$
$439$	−30.3050	−1.44638	−0.723188	+	0.690651i	$0.757324\pi$
$440$	0	0
$441$	−28.6525	−1.36440
$442$	0	0
$443$	−37.1803	−1.76649	−0.883246	−	0.468911i	$-0.844647\pi$
$443$	−37.1803	−1.76649	−0.883246	+	0.468911i	$0.844647\pi$
$444$	0	0
$445$	8.29180	0.393069
$446$	0	0
$447$	8.27051	0.391182
$448$	0	0
$449$	8.88854	0.419476	0.209738	−	0.977758i	$-0.432739\pi$
$449$	8.88854	0.419476	0.209738	+	0.977758i	$0.432739\pi$
$450$	0	0
$451$	65.8115	3.09894
$452$	0	0
$453$	−12.4164	−0.583374
$454$	0	0
$455$	−28.6525	−1.34325
$456$	0	0
$457$	−29.1246	−1.36239	−0.681196	−	0.732101i	$-0.738540\pi$
$457$	−29.1246	−1.36239	−0.681196	+	0.732101i	$0.738540\pi$
$458$	0	0
$459$	−9.59675	−0.447938
$460$	0	0
$461$	−30.8328	−1.43603	−0.718014	−	0.696029i	$-0.754949\pi$
$461$	−30.8328	−1.43603	−0.718014	+	0.696029i	$0.754949\pi$
$462$	0	0
$463$	−8.50658	−0.395334	−0.197667	−	0.980269i	$-0.563337\pi$
$463$	−8.50658	−0.395334	−0.197667	+	0.980269i	$0.563337\pi$
$464$	0	0
$465$	−5.05573	−0.234454
$466$	0	0
$467$	14.2361	0.658767	0.329383	−	0.944196i	$-0.393159\pi$
$467$	14.2361	0.658767	0.329383	+	0.944196i	$0.393159\pi$
$468$	0	0
$469$	28.4164	1.31215
$470$	0	0
$471$	8.52786	0.392943
$472$	0	0
$473$	−10.7082	−0.492364
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.47214	0.342126
$478$	0	0
$479$	−11.6180	−0.530842	−0.265421	−	0.964133i	$-0.585511\pi$
$479$	−11.6180	−0.530842	−0.265421	+	0.964133i	$0.585511\pi$
$480$	0	0
$481$	47.1591	2.15027
$482$	0	0
$483$	−2.23607	−0.101745
$484$	0	0
$485$	−0.832816	−0.0378162
$486$	0	0
$487$	9.23607	0.418526	0.209263	−	0.977859i	$-0.432893\pi$
$487$	9.23607	0.418526	0.209263	+	0.977859i	$0.432893\pi$
$488$	0	0
$489$	−8.90983	−0.402916
$490$	0	0
$491$	−19.2705	−0.869666	−0.434833	−	0.900511i	$-0.643193\pi$
$491$	−19.2705	−0.869666	−0.434833	+	0.900511i	$0.643193\pi$
$492$	0	0
$493$	10.6525	0.479763
$494$	0	0
$495$	21.4164	0.962596
$496$	0	0
$497$	30.4164	1.36436
$498$	0	0
$499$	−30.1803	−1.35106	−0.675529	−	0.737334i	$-0.736085\pi$
$499$	−30.1803	−1.35106	−0.675529	+	0.737334i	$0.736085\pi$
$500$	0	0
$501$	7.74265	0.345916
$502$	0	0
$503$	19.7639	0.881230	0.440615	−	0.897696i	$-0.354760\pi$
$503$	19.7639	0.881230	0.440615	+	0.897696i	$0.354760\pi$
$504$	0	0
$505$	5.23607	0.233002
$506$	0	0
$507$	10.4721	0.465084
$508$	0	0
$509$	41.8673	1.85573	0.927867	−	0.372912i	$-0.121641\pi$
$509$	41.8673	1.85573	0.927867	+	0.372912i	$0.121641\pi$
$510$	0	0
$511$	51.8328	2.29295
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	4.68692	0.206130
$518$	0	0
$519$	10.1803	0.446867
$520$	0	0
$521$	−8.14590	−0.356878	−0.178439	−	0.983951i	$-0.557105\pi$
$521$	−8.14590	−0.356878	−0.178439	+	0.983951i	$0.557105\pi$
$522$	0	0
$523$	−35.0689	−1.53346	−0.766728	−	0.641973i	$-0.778116\pi$
$523$	−35.0689	−1.53346	−0.766728	+	0.641973i	$0.778116\pi$
$524$	0	0
$525$	−9.09017	−0.396728
$526$	0	0
$527$	−18.2918	−0.796803
$528$	0	0
$529$	−22.2705	−0.968283
$530$	0	0
$531$	19.9443	0.865508
$532$	0	0
$533$	54.4164	2.35704
$534$	0	0
$535$	4.06888	0.175913
$536$	0	0
$537$	−10.5066	−0.453392
$538$	0	0
$539$	−72.4296	−3.11976
$540$	0	0
$541$	−34.4164	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$541$	−34.4164	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$542$	0	0
$543$	−7.41641	−0.318269
$544$	0	0
$545$	−5.23607	−0.224289
$546$	0	0
$547$	7.56231	0.323341	0.161670	−	0.986845i	$-0.448312\pi$
$547$	7.56231	0.323341	0.161670	+	0.986845i	$0.448312\pi$
$548$	0	0
$549$	17.5623	0.749541
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−25.4164	−1.08082
$554$	0	0
$555$	−6.58359	−0.279458
$556$	0	0
$557$	−23.1803	−0.982183	−0.491091	−	0.871108i	$-0.663402\pi$
$557$	−23.1803	−0.982183	−0.491091	+	0.871108i	$0.663402\pi$
$558$	0	0
$559$	−8.85410	−0.374489
$560$	0	0
$561$	−11.3050	−0.477295
$562$	0	0
$563$	−10.5836	−0.446045	−0.223023	−	0.974813i	$-0.571592\pi$
$563$	−10.5836	−0.446045	−0.223023	+	0.974813i	$0.571592\pi$
$564$	0	0
$565$	7.77709	0.327185
$566$	0	0
$567$	24.1803	1.01548
$568$	0	0
$569$	−10.5623	−0.442795	−0.221397	−	0.975184i	$-0.571062\pi$
$569$	−10.5623	−0.442795	−0.221397	+	0.975184i	$0.571062\pi$
$570$	0	0
$571$	−8.32624	−0.348442	−0.174221	−	0.984707i	$-0.555741\pi$
$571$	−8.32624	−0.348442	−0.174221	+	0.984707i	$0.555741\pi$
$572$	0	0
$573$	2.43769	0.101836
$574$	0	0
$575$	2.96556	0.123672
$576$	0	0
$577$	−24.2361	−1.00896	−0.504480	−	0.863423i	$-0.668316\pi$
$577$	−24.2361	−1.00896	−0.504480	+	0.863423i	$0.668316\pi$
$578$	0	0
$579$	−3.70820	−0.154108
$580$	0	0
$581$	12.4721	0.517431
$582$	0	0
$583$	18.8885	0.782284
$584$	0	0
$585$	17.7082	0.732144
$586$	0	0
$587$	17.8328	0.736039	0.368020	−	0.929818i	$-0.380036\pi$
$587$	17.8328	0.736039	0.368020	+	0.929818i	$0.380036\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−7.09017	−0.291651
$592$	0	0
$593$	−1.29180	−0.0530477	−0.0265239	−	0.999648i	$-0.508444\pi$
$593$	−1.29180	−0.0530477	−0.0265239	+	0.999648i	$0.508444\pi$
$594$	0	0
$595$	14.4721	0.593300
$596$	0	0
$597$	−11.7082	−0.479185
$598$	0	0
$599$	−28.1246	−1.14914	−0.574570	−	0.818455i	$-0.694831\pi$
$599$	−28.1246	−1.14914	−0.574570	+	0.818455i	$0.694831\pi$
$600$	0	0
$601$	−24.1803	−0.986337	−0.493168	−	0.869934i	$-0.664161\pi$
$601$	−24.1803	−0.986337	−0.493168	+	0.869934i	$0.664161\pi$
$602$	0	0
$603$	−17.5623	−0.715192
$604$	0	0
$605$	40.5410	1.64823
$606$	0	0
$607$	31.3820	1.27375	0.636877	−	0.770965i	$-0.280226\pi$
$607$	31.3820	1.27375	0.636877	+	0.770965i	$0.280226\pi$
$608$	0	0
$609$	10.0902	0.408874
$610$	0	0
$611$	3.87539	0.156781
$612$	0	0
$613$	30.3050	1.22401	0.612003	−	0.790856i	$-0.290364\pi$
$613$	30.3050	1.22401	0.612003	+	0.790856i	$0.290364\pi$
$614$	0	0
$615$	−7.59675	−0.306330
$616$	0	0
$617$	5.43769	0.218913	0.109457	−	0.993992i	$-0.465089\pi$
$617$	5.43769	0.218913	0.109457	+	0.993992i	$0.465089\pi$
$618$	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
$621$	2.96556	0.119004
$622$	0	0
$623$	28.4164	1.13848
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−23.8197	−0.949752
$630$	0	0
$631$	−33.5410	−1.33525	−0.667623	−	0.744499i	$-0.732688\pi$
$631$	−33.5410	−1.33525	−0.667623	+	0.744499i	$0.732688\pi$
$632$	0	0
$633$	2.52786	0.100474
$634$	0	0
$635$	−11.1246	−0.441467
$636$	0	0
$637$	−59.8885	−2.37287
$638$	0	0
$639$	−18.7984	−0.743652
$640$	0	0
$641$	−4.61803	−0.182401	−0.0912007	−	0.995833i	$-0.529070\pi$
$641$	−4.61803	−0.182401	−0.0912007	+	0.995833i	$0.529070\pi$
$642$	0	0
$643$	−24.1803	−0.953580	−0.476790	−	0.879017i	$-0.658200\pi$
$643$	−24.1803	−0.953580	−0.476790	+	0.879017i	$0.658200\pi$
$644$	0	0
$645$	1.23607	0.0486701
$646$	0	0
$647$	33.0689	1.30007	0.650036	−	0.759903i	$-0.274754\pi$
$647$	33.0689	1.30007	0.650036	+	0.759903i	$0.274754\pi$
$648$	0	0
$649$	50.4164	1.97902
$650$	0	0
$651$	−17.3262	−0.679069
$652$	0	0
$653$	−3.79837	−0.148642	−0.0743209	−	0.997234i	$-0.523679\pi$
$653$	−3.79837	−0.148642	−0.0743209	+	0.997234i	$0.523679\pi$
$654$	0	0
$655$	−9.41641	−0.367930
$656$	0	0
$657$	−32.0344	−1.24978
$658$	0	0
$659$	−18.3607	−0.715231	−0.357615	−	0.933869i	$-0.616410\pi$
$659$	−18.3607	−0.715231	−0.357615	+	0.933869i	$0.616410\pi$
$660$	0	0
$661$	9.41641	0.366256	0.183128	−	0.983089i	$-0.441378\pi$
$661$	9.41641	0.366256	0.183128	+	0.983089i	$0.441378\pi$
$662$	0	0
$663$	−9.34752	−0.363028
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.29180	−0.127459
$668$	0	0
$669$	−4.32624	−0.167262
$670$	0	0
$671$	44.3951	1.71385
$672$	0	0
$673$	48.5967	1.87327	0.936633	−	0.350311i	$-0.113924\pi$
$673$	48.5967	1.87327	0.936633	+	0.350311i	$0.113924\pi$
$674$	0	0
$675$	12.0557	0.464025
$676$	0	0
$677$	14.5066	0.557533	0.278767	−	0.960359i	$-0.410074\pi$
$677$	14.5066	0.557533	0.278767	+	0.960359i	$0.410074\pi$
$678$	0	0
$679$	−2.85410	−0.109530
$680$	0	0
$681$	−10.3262	−0.395702
$682$	0	0
$683$	21.9443	0.839674	0.419837	−	0.907599i	$-0.362087\pi$
$683$	21.9443	0.839674	0.419837	+	0.907599i	$0.362087\pi$
$684$	0	0
$685$	19.1246	0.730714
$686$	0	0
$687$	0.742646	0.0283337
$688$	0	0
$689$	15.6180	0.595000
$690$	0	0
$691$	−0.527864	−0.0200809	−0.0100404	−	0.999950i	$-0.503196\pi$
$691$	−0.527864	−0.0200809	−0.0100404	+	0.999950i	$0.503196\pi$
$692$	0	0
$693$	73.3951	2.78805
$694$	0	0
$695$	−13.4164	−0.508913
$696$	0	0
$697$	−27.4853	−1.04108
$698$	0	0
$699$	13.2705	0.501937
$700$	0	0
$701$	−0.201626	−0.00761531	−0.00380766	−	0.999993i	$-0.501212\pi$
$701$	−0.201626	−0.00761531	−0.00380766	+	0.999993i	$0.501212\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.541020	−0.0203760
$706$	0	0
$707$	17.9443	0.674864
$708$	0	0
$709$	16.3607	0.614438	0.307219	−	0.951639i	$-0.400602\pi$
$709$	16.3607	0.614438	0.307219	+	0.951639i	$0.400602\pi$
$710$	0	0
$711$	15.7082	0.589104
$712$	0	0
$713$	5.65248	0.211687
$714$	0	0
$715$	44.7639	1.67408
$716$	0	0
$717$	3.87539	0.144729
$718$	0	0
$719$	−36.3262	−1.35474	−0.677370	−	0.735642i	$-0.736880\pi$
$719$	−36.3262	−1.35474	−0.677370	+	0.735642i	$0.736880\pi$
$720$	0	0
$721$	20.5623	0.765780
$722$	0	0
$723$	−12.5066	−0.465125
$724$	0	0
$725$	−13.3820	−0.496994
$726$	0	0
$727$	24.7639	0.918443	0.459222	−	0.888322i	$-0.348128\pi$
$727$	24.7639	0.918443	0.459222	+	0.888322i	$0.348128\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	4.47214	0.165408
$732$	0	0
$733$	−9.65248	−0.356522	−0.178261	−	0.983983i	$-0.557047\pi$
$733$	−9.65248	−0.356522	−0.178261	+	0.983983i	$0.557047\pi$
$734$	0	0
$735$	8.36068	0.308388
$736$	0	0
$737$	−44.3951	−1.63532
$738$	0	0
$739$	−9.06888	−0.333604	−0.166802	−	0.985990i	$-0.553344\pi$
$739$	−9.06888	−0.333604	−0.166802	+	0.985990i	$0.553344\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.7082	1.56681	0.783406	−	0.621510i	$-0.213481\pi$
$743$	42.7082	1.56681	0.783406	+	0.621510i	$0.213481\pi$
$744$	0	0
$745$	16.5410	0.606016
$746$	0	0
$747$	−7.70820	−0.282028
$748$	0	0
$749$	13.9443	0.509513
$750$	0	0
$751$	23.0902	0.842572	0.421286	−	0.906928i	$-0.361579\pi$
$751$	23.0902	0.842572	0.421286	+	0.906928i	$0.361579\pi$
$752$	0	0
$753$	−9.74265	−0.355042
$754$	0	0
$755$	−24.8328	−0.903759
$756$	0	0
$757$	−46.7984	−1.70092	−0.850458	−	0.526043i	$-0.823675\pi$
$757$	−46.7984	−1.70092	−0.850458	+	0.526043i	$0.823675\pi$
$758$	0	0
$759$	3.49342	0.126803
$760$	0	0
$761$	49.8328	1.80644	0.903219	−	0.429180i	$-0.141197\pi$
$761$	49.8328	1.80644	0.903219	+	0.429180i	$0.141197\pi$
$762$	0	0
$763$	−17.9443	−0.649626
$764$	0	0
$765$	−8.94427	−0.323381
$766$	0	0
$767$	41.6869	1.50523
$768$	0	0
$769$	−27.3820	−0.987419	−0.493709	−	0.869627i	$-0.664359\pi$
$769$	−27.3820	−0.987419	−0.493709	+	0.869627i	$0.664359\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	0	0
$773$	3.43769	0.123645	0.0618226	−	0.998087i	$-0.480309\pi$
$773$	3.43769	0.123645	0.0618226	+	0.998087i	$0.480309\pi$
$774$	0	0
$775$	22.9787	0.825420
$776$	0	0
$777$	−22.5623	−0.809418
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−47.5197	−1.70039
$782$	0	0
$783$	−13.3820	−0.478232
$784$	0	0
$785$	17.0557	0.608745
$786$	0	0
$787$	−43.3050	−1.54365	−0.771827	−	0.635832i	$-0.780657\pi$
$787$	−43.3050	−1.54365	−0.771827	+	0.635832i	$0.780657\pi$
$788$	0	0
$789$	2.76393	0.0983986
$790$	0	0
$791$	26.6525	0.947653
$792$	0	0
$793$	36.7082	1.30355
$794$	0	0
$795$	−2.18034	−0.0773287
$796$	0	0
$797$	34.0689	1.20678	0.603391	−	0.797446i	$-0.293816\pi$
$797$	34.0689	1.20678	0.603391	+	0.797446i	$0.293816\pi$
$798$	0	0
$799$	−1.95743	−0.0692488
$800$	0	0
$801$	−17.5623	−0.620534
$802$	0	0
$803$	−80.9787	−2.85768
$804$	0	0
$805$	−4.47214	−0.157622
$806$	0	0
$807$	0.819660	0.0288534
$808$	0	0
$809$	−1.50658	−0.0529685	−0.0264842	−	0.999649i	$-0.508431\pi$
$809$	−1.50658	−0.0529685	−0.0264842	+	0.999649i	$0.508431\pi$
$810$	0	0
$811$	1.25735	0.0441517	0.0220758	−	0.999756i	$-0.492972\pi$
$811$	1.25735	0.0441517	0.0220758	+	0.999756i	$0.492972\pi$
$812$	0	0
$813$	−11.9787	−0.420112
$814$	0	0
$815$	−17.8197	−0.624195
$816$	0	0
$817$	0	0
$818$	0	0
$819$	60.6869	2.12057
$820$	0	0
$821$	0.257354	0.00898172	0.00449086	−	0.999990i	$-0.498571\pi$
$821$	0.257354	0.00898172	0.00449086	+	0.999990i	$0.498571\pi$
$822$	0	0
$823$	19.2361	0.670527	0.335264	−	0.942124i	$-0.391175\pi$
$823$	19.2361	0.670527	0.335264	+	0.942124i	$0.391175\pi$
$824$	0	0
$825$	14.2016	0.494437
$826$	0	0
$827$	−36.4164	−1.26632	−0.633161	−	0.774020i	$-0.718243\pi$
$827$	−36.4164	−1.26632	−0.633161	+	0.774020i	$0.718243\pi$
$828$	0	0
$829$	33.5623	1.16567	0.582834	−	0.812592i	$-0.301944\pi$
$829$	33.5623	1.16567	0.582834	+	0.812592i	$0.301944\pi$
$830$	0	0
$831$	4.58359	0.159003
$832$	0	0
$833$	30.2492	1.04807
$834$	0	0
$835$	15.4853	0.535891
$836$	0	0
$837$	22.9787	0.794261
$838$	0	0
$839$	14.9230	0.515199	0.257599	−	0.966252i	$-0.417069\pi$
$839$	14.9230	0.515199	0.257599	+	0.966252i	$0.417069\pi$
$840$	0	0
$841$	−14.1459	−0.487790
$842$	0	0
$843$	−8.09017	−0.278640
$844$	0	0
$845$	20.9443	0.720505
$846$	0	0
$847$	138.936	4.77390
$848$	0	0
$849$	−9.00000	−0.308879
$850$	0	0
$851$	7.36068	0.252321
$852$	0	0
$853$	11.1803	0.382808	0.191404	−	0.981511i	$-0.438696\pi$
$853$	11.1803	0.382808	0.191404	+	0.981511i	$0.438696\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	58.0476	1.98287	0.991434	−	0.130608i	$-0.0416930\pi$
$857$	58.0476	1.98287	0.991434	+	0.130608i	$0.0416930\pi$
$858$	0	0
$859$	30.5410	1.04205	0.521023	−	0.853543i	$-0.325551\pi$
$859$	30.5410	1.04205	0.521023	+	0.853543i	$0.325551\pi$
$860$	0	0
$861$	−26.0344	−0.887251
$862$	0	0
$863$	35.7771	1.21787	0.608933	−	0.793222i	$-0.291598\pi$
$863$	35.7771	1.21787	0.608933	+	0.793222i	$0.291598\pi$
$864$	0	0
$865$	20.3607	0.692284
$866$	0	0
$867$	−5.78522	−0.196476
$868$	0	0
$869$	39.7082	1.34701
$870$	0	0
$871$	−36.7082	−1.24381
$872$	0	0
$873$	1.76393	0.0597001
$874$	0	0
$875$	−44.3607	−1.49966
$876$	0	0
$877$	20.0689	0.677678	0.338839	−	0.940844i	$-0.389966\pi$
$877$	20.0689	0.677678	0.338839	+	0.940844i	$0.389966\pi$
$878$	0	0
$879$	3.00000	0.101187
$880$	0	0
$881$	−4.25735	−0.143434	−0.0717170	−	0.997425i	$-0.522848\pi$
$881$	−4.25735	−0.143434	−0.0717170	+	0.997425i	$0.522848\pi$
$882$	0	0
$883$	−19.2016	−0.646186	−0.323093	−	0.946367i	$-0.604723\pi$
$883$	−19.2016	−0.646186	−0.323093	+	0.946367i	$0.604723\pi$
$884$	0	0
$885$	−5.81966	−0.195626
$886$	0	0
$887$	−6.29180	−0.211258	−0.105629	−	0.994406i	$-0.533686\pi$
$887$	−6.29180	−0.211258	−0.105629	+	0.994406i	$0.533686\pi$
$888$	0	0
$889$	−38.1246	−1.27866
$890$	0	0
$891$	−37.7771	−1.26558
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−21.0132	−0.702392
$896$	0	0
$897$	2.88854	0.0964457
$898$	0	0
$899$	−25.5066	−0.850692
$900$	0	0
$901$	−7.88854	−0.262805
$902$	0	0
$903$	4.23607	0.140968
$904$	0	0
$905$	−14.8328	−0.493059
$906$	0	0
$907$	−20.5836	−0.683467	−0.341733	−	0.939797i	$-0.611014\pi$
$907$	−20.5836	−0.683467	−0.341733	+	0.939797i	$0.611014\pi$
$908$	0	0
$909$	−11.0902	−0.367838
$910$	0	0
$911$	−47.3951	−1.57027	−0.785135	−	0.619324i	$-0.787407\pi$
$911$	−47.3951	−1.57027	−0.785135	+	0.619324i	$0.787407\pi$
$912$	0	0
$913$	−19.4853	−0.644869
$914$	0	0
$915$	−5.12461	−0.169414
$916$	0	0
$917$	−32.2705	−1.06567
$918$	0	0
$919$	39.2492	1.29471	0.647356	−	0.762188i	$-0.275875\pi$
$919$	39.2492	1.29471	0.647356	+	0.762188i	$0.275875\pi$
$920$	0	0
$921$	4.78522	0.157678
$922$	0	0
$923$	−39.2918	−1.29331
$924$	0	0
$925$	29.9230	0.983862
$926$	0	0
$927$	−12.7082	−0.417392
$928$	0	0
$929$	0.506578	0.0166203	0.00831014	−	0.999965i	$-0.497355\pi$
$929$	0.506578	0.0166203	0.00831014	+	0.999965i	$0.497355\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	10.9443	0.358299
$934$	0	0
$935$	−22.6099	−0.739423
$936$	0	0
$937$	25.1591	0.821910	0.410955	−	0.911656i	$-0.365195\pi$
$937$	25.1591	0.821910	0.410955	+	0.911656i	$0.365195\pi$
$938$	0	0
$939$	9.50658	0.310235
$940$	0	0
$941$	40.9230	1.33405	0.667026	−	0.745035i	$-0.267567\pi$
$941$	40.9230	1.33405	0.667026	+	0.745035i	$0.267567\pi$
$942$	0	0
$943$	8.49342	0.276584
$944$	0	0
$945$	−18.1803	−0.591407
$946$	0	0
$947$	43.0132	1.39774	0.698870	−	0.715249i	$-0.253687\pi$
$947$	43.0132	1.39774	0.698870	+	0.715249i	$0.253687\pi$
$948$	0	0
$949$	−66.9574	−2.17353
$950$	0	0
$951$	−6.76393	−0.219336
$952$	0	0
$953$	−27.2918	−0.884068	−0.442034	−	0.896998i	$-0.645743\pi$
$953$	−27.2918	−0.884068	−0.442034	+	0.896998i	$0.645743\pi$
$954$	0	0
$955$	4.87539	0.157764
$956$	0	0
$957$	−15.7639	−0.509576
$958$	0	0
$959$	65.5410	2.11643
$960$	0	0
$961$	12.7984	0.412851
$962$	0	0
$963$	−8.61803	−0.277712
$964$	0	0
$965$	−7.41641	−0.238743
$966$	0	0
$967$	22.6525	0.728455	0.364227	−	0.931310i	$-0.381333\pi$
$967$	22.6525	0.728455	0.364227	+	0.931310i	$0.381333\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.4377	1.04098	0.520488	−	0.853869i	$-0.325750\pi$
$971$	32.4377	1.04098	0.520488	+	0.853869i	$0.325750\pi$
$972$	0	0
$973$	−45.9787	−1.47401
$974$	0	0
$975$	11.7426	0.376066
$976$	0	0
$977$	9.58359	0.306606	0.153303	−	0.988179i	$-0.451009\pi$
$977$	9.58359	0.306606	0.153303	+	0.988179i	$0.451009\pi$
$978$	0	0
$979$	−44.3951	−1.41887
$980$	0	0
$981$	11.0902	0.354082
$982$	0	0
$983$	−5.32624	−0.169881	−0.0849403	−	0.996386i	$-0.527070\pi$
$983$	−5.32624	−0.169881	−0.0849403	+	0.996386i	$0.527070\pi$
$984$	0	0
$985$	−14.1803	−0.451823
$986$	0	0
$987$	−1.85410	−0.0590167
$988$	0	0
$989$	−1.38197	−0.0439440
$990$	0	0
$991$	20.2016	0.641726	0.320863	−	0.947126i	$-0.396027\pi$
$991$	20.2016	0.641726	0.320863	+	0.947126i	$0.396027\pi$
$992$	0	0
$993$	−4.65248	−0.147642
$994$	0	0
$995$	−23.4164	−0.742350
$996$	0	0
$997$	−10.2918	−0.325944	−0.162972	−	0.986631i	$-0.552108\pi$
$997$	−10.2918	−0.325944	−0.162972	+	0.986631i	$0.552108\pi$
$998$	0	0
$999$	29.9230	0.946721

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} +1.23607 q^{5} +4.23607 q^{7} -2.61803 q^{9} -6.61803 q^{11} -5.47214 q^{13} +0.763932 q^{15} +2.76393 q^{17} +2.61803 q^{21} -0.854102 q^{23} -3.47214 q^{25} -3.47214 q^{27} +3.85410 q^{29} -6.61803 q^{31} -4.09017 q^{33} +5.23607 q^{35} -8.61803 q^{37} -3.38197 q^{39} -9.94427 q^{41} +1.61803 q^{43} -3.23607 q^{45} -0.708204 q^{47} +10.9443 q^{49} +1.70820 q^{51} -2.85410 q^{53} -8.18034 q^{55} -7.61803 q^{59} -6.70820 q^{61} -11.0902 q^{63} -6.76393 q^{65} +6.70820 q^{67} -0.527864 q^{69} +7.18034 q^{71} +12.2361 q^{73} -2.14590 q^{75} -28.0344 q^{77} -6.00000 q^{79} +5.70820 q^{81} +2.94427 q^{83} +3.41641 q^{85} +2.38197 q^{87} +6.70820 q^{89} -23.1803 q^{91} -4.09017 q^{93} -0.673762 q^{97} +17.3262 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 4 q^{7} - 3 q^{9} - 11 q^{11} - 2 q^{13} + 6 q^{15} + 10 q^{17} + 3 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + q^{29} - 11 q^{31} + 3 q^{33} + 6 q^{35} - 15 q^{37} - 9 q^{39} - 2 q^{41}+ \cdots + 19 q^{99}+O(q^{100}) \) 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

Properties

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