LMFDB - Embedded newform 6664.2.a.o.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.10522$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.10522 q^{3} +3.40632 q^{5} +1.43195 q^{9} -1.23607 q^{11} +6.47214 q^{13} +7.17104 q^{15} -1.00000 q^{17} +2.21044 q^{19} +1.39781 q^{23} +6.60299 q^{25} -3.30110 q^{27} -0.633874 q^{29} -0.965861 q^{31} -2.60219 q^{33} -8.31040 q^{37} +13.6253 q^{39} +5.60299 q^{41} +11.0796 q^{43} +4.87766 q^{45} -7.67652 q^{47} -2.10522 q^{51} +11.7167 q^{53} -4.21044 q^{55} +4.65345 q^{57} +0.602193 q^{59} +9.84352 q^{61} +22.0461 q^{65} +5.30636 q^{67} +2.94269 q^{69} -3.33603 q^{71} -14.0239 q^{73} +13.9007 q^{75} -0.323477 q^{79} -11.2454 q^{81} -13.8870 q^{83} -3.40632 q^{85} -1.33444 q^{87} +14.7509 q^{89} -2.03335 q^{93} +7.52945 q^{95} +5.90855 q^{97} -1.76998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} + 7 q^{9} + 4 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{17} - 14 q^{19} + 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} - 8 q^{33} - 4 q^{37} + 4 q^{39} + 7 q^{41} + 19 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 + 7 * q^9 + 4 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^17 - 14 * q^19 + 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 - 8 * q^33 - 4 * q^37 + 4 * q^39 + 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 - 8 * q^65 + 15 * q^67 + 2 * q^69 + 2 * q^71 + 5 * q^73 - 10 * q^75 - 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 + 16 * q^89 - 20 * q^93 + 22 * q^95 + 15 * q^97 + 2 * 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
$3$	2.10522	1.21545	0.607724	−	0.794148i	$-0.292083\pi$
$3$	2.10522	1.21545	0.607724	+	0.794148i	$0.292083\pi$
$4$	0	0
$5$	3.40632	1.52335	0.761675	−	0.647959i	$-0.224377\pi$
$5$	3.40632	1.52335	0.761675	+	0.647959i	$0.224377\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.43195	0.477315
$10$	0	0
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	6.47214	1.79505	0.897524	−	0.440966i	$-0.145364\pi$
$13$	6.47214	1.79505	0.897524	+	0.440966i	$0.145364\pi$
$14$	0	0
$15$	7.17104	1.85155
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.21044	0.507109	0.253555	−	0.967321i	$-0.418400\pi$
$19$	2.21044	0.507109	0.253555	+	0.967321i	$0.418400\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.39781	0.291463	0.145731	−	0.989324i	$-0.453446\pi$
$23$	1.39781	0.291463	0.145731	+	0.989324i	$0.453446\pi$
$24$	0	0
$25$	6.60299	1.32060
$26$	0	0
$27$	−3.30110	−0.635296
$28$	0	0
$29$	−0.633874	−0.117708	−0.0588538	−	0.998267i	$-0.518745\pi$
$29$	−0.633874	−0.117708	−0.0588538	+	0.998267i	$0.518745\pi$
$30$	0	0
$31$	−0.965861	−0.173474	−0.0867368	−	0.996231i	$-0.527644\pi$
$31$	−0.965861	−0.173474	−0.0867368	+	0.996231i	$0.527644\pi$
$32$	0	0
$33$	−2.60219	−0.452984
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.31040	−1.36622	−0.683110	−	0.730315i	$-0.739373\pi$
$37$	−8.31040	−1.36622	−0.683110	+	0.730315i	$0.739373\pi$
$38$	0	0
$39$	13.6253	2.18179
$40$	0	0
$41$	5.60299	0.875039	0.437520	−	0.899209i	$-0.355857\pi$
$41$	5.60299	0.875039	0.437520	+	0.899209i	$0.355857\pi$
$42$	0	0
$43$	11.0796	1.68962	0.844811	−	0.535065i	$-0.179713\pi$
$43$	11.0796	1.68962	0.844811	+	0.535065i	$0.179713\pi$
$44$	0	0
$45$	4.87766	0.727119
$46$	0	0
$47$	−7.67652	−1.11974	−0.559868	−	0.828582i	$-0.689148\pi$
$47$	−7.67652	−1.11974	−0.559868	+	0.828582i	$0.689148\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.10522	−0.294790
$52$	0	0
$53$	11.7167	1.60941	0.804707	−	0.593672i	$-0.202322\pi$
$53$	11.7167	1.60941	0.804707	+	0.593672i	$0.202322\pi$
$54$	0	0
$55$	−4.21044	−0.567735
$56$	0	0
$57$	4.65345	0.616365
$58$	0	0
$59$	0.602193	0.0783989	0.0391995	−	0.999231i	$-0.487519\pi$
$59$	0.602193	0.0783989	0.0391995	+	0.999231i	$0.487519\pi$
$60$	0	0
$61$	9.84352	1.26033	0.630167	−	0.776460i	$-0.282986\pi$
$61$	9.84352	1.26033	0.630167	+	0.776460i	$0.282986\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.0461	2.73449
$66$	0	0
$67$	5.30636	0.648275	0.324137	−	0.946010i	$-0.394926\pi$
$67$	5.30636	0.648275	0.324137	+	0.946010i	$0.394926\pi$
$68$	0	0
$69$	2.94269	0.354258
$70$	0	0
$71$	−3.33603	−0.395914	−0.197957	−	0.980211i	$-0.563431\pi$
$71$	−3.33603	−0.395914	−0.197957	+	0.980211i	$0.563431\pi$
$72$	0	0
$73$	−14.0239	−1.64137	−0.820684	−	0.571382i	$-0.806408\pi$
$73$	−14.0239	−1.64137	−0.820684	+	0.571382i	$0.806408\pi$
$74$	0	0
$75$	13.9007	1.60512
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.323477	−0.0363940	−0.0181970	−	0.999834i	$-0.505793\pi$
$79$	−0.323477	−0.0363940	−0.0181970	+	0.999834i	$0.505793\pi$
$80$	0	0
$81$	−11.2454	−1.24949
$82$	0	0
$83$	−13.8870	−1.52429	−0.762146	−	0.647405i	$-0.775854\pi$
$83$	−13.8870	−1.52429	−0.762146	+	0.647405i	$0.775854\pi$
$84$	0	0
$85$	−3.40632	−0.369467
$86$	0	0
$87$	−1.33444	−0.143067
$88$	0	0
$89$	14.7509	1.56359	0.781794	−	0.623537i	$-0.214305\pi$
$89$	14.7509	1.56359	0.781794	+	0.623537i	$0.214305\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.03335	−0.210848
$94$	0	0
$95$	7.52945	0.772505
$96$	0	0
$97$	5.90855	0.599922	0.299961	−	0.953951i	$-0.403026\pi$
$97$	5.90855	0.599922	0.299961	+	0.953951i	$0.403026\pi$
$98$	0	0
$99$	−1.76998	−0.177890
$100$	0	0
$101$	−5.08485	−0.505961	−0.252981	−	0.967471i	$-0.581411\pi$
$101$	−5.08485	−0.505961	−0.252981	+	0.967471i	$0.581411\pi$
$102$	0	0
$103$	5.28477	0.520724	0.260362	−	0.965511i	$-0.416158\pi$
$103$	5.28477	0.520724	0.260362	+	0.965511i	$0.416158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.25309	−0.894530	−0.447265	−	0.894402i	$-0.647602\pi$
$107$	−9.25309	−0.894530	−0.447265	+	0.894402i	$0.647602\pi$
$108$	0	0
$109$	−16.1000	−1.54210	−0.771048	−	0.636777i	$-0.780267\pi$
$109$	−16.1000	−1.54210	−0.771048	+	0.636777i	$0.780267\pi$
$110$	0	0
$111$	−17.4952	−1.66057
$112$	0	0
$113$	10.5509	0.992548	0.496274	−	0.868166i	$-0.334701\pi$
$113$	10.5509	0.992548	0.496274	+	0.868166i	$0.334701\pi$
$114$	0	0
$115$	4.76137	0.444000
$116$	0	0
$117$	9.26775	0.856804
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	11.7955	1.06357
$124$	0	0
$125$	5.46027	0.488382
$126$	0	0
$127$	−17.9272	−1.59078	−0.795389	−	0.606100i	$-0.792733\pi$
$127$	−17.9272	−1.59078	−0.795389	+	0.606100i	$0.792733\pi$
$128$	0	0
$129$	23.3250	2.05365
$130$	0	0
$131$	−1.49777	−0.130860	−0.0654302	−	0.997857i	$-0.520842\pi$
$131$	−1.49777	−0.130860	−0.0654302	+	0.997857i	$0.520842\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.2446	−0.967779
$136$	0	0
$137$	13.6030	1.16218	0.581091	−	0.813839i	$-0.302626\pi$
$137$	13.6030	1.16218	0.581091	+	0.813839i	$0.302626\pi$
$138$	0	0
$139$	6.28073	0.532724	0.266362	−	0.963873i	$-0.414178\pi$
$139$	6.28073	0.532724	0.266362	+	0.963873i	$0.414178\pi$
$140$	0	0
$141$	−16.1608	−1.36098
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.15918	−0.179310
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.7004	1.77776	0.888882	−	0.458136i	$-0.151483\pi$
$149$	21.7004	1.77776	0.888882	+	0.458136i	$0.151483\pi$
$150$	0	0
$151$	4.60745	0.374949	0.187475	−	0.982269i	$-0.439970\pi$
$151$	4.60745	0.374949	0.187475	+	0.982269i	$0.439970\pi$
$152$	0	0
$153$	−1.43195	−0.115766
$154$	0	0
$155$	−3.29003	−0.264261
$156$	0	0
$157$	4.53391	0.361846	0.180923	−	0.983497i	$-0.442092\pi$
$157$	4.53391	0.361846	0.180923	+	0.983497i	$0.442092\pi$
$158$	0	0
$159$	24.6662	1.95616
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.2421	−0.958877	−0.479438	−	0.877576i	$-0.659160\pi$
$163$	−12.2421	−0.958877	−0.479438	+	0.877576i	$0.659160\pi$
$164$	0	0
$165$	−8.86389	−0.690053
$166$	0	0
$167$	−1.34129	−0.103792	−0.0518959	−	0.998652i	$-0.516526\pi$
$167$	−1.34129	−0.103792	−0.0518959	+	0.998652i	$0.516526\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	0	0
$171$	3.16523	0.242051
$172$	0	0
$173$	−1.83300	−0.139361	−0.0696803	−	0.997569i	$-0.522198\pi$
$173$	−1.83300	−0.139361	−0.0696803	+	0.997569i	$0.522198\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.26775	0.0952898
$178$	0	0
$179$	20.2343	1.51238	0.756191	−	0.654351i	$-0.227058\pi$
$179$	20.2343	1.51238	0.756191	+	0.654351i	$0.227058\pi$
$180$	0	0
$181$	6.03010	0.448214	0.224107	−	0.974565i	$-0.428054\pi$
$181$	6.03010	0.448214	0.224107	+	0.974565i	$0.428054\pi$
$182$	0	0
$183$	20.7228	1.53187
$184$	0	0
$185$	−28.3078	−2.08123
$186$	0	0
$187$	1.23607	0.0903902
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.9146	−0.789753	−0.394876	−	0.918734i	$-0.629213\pi$
$191$	−10.9146	−0.789753	−0.394876	+	0.918734i	$0.629213\pi$
$192$	0	0
$193$	−19.1717	−1.38001	−0.690006	−	0.723804i	$-0.742392\pi$
$193$	−19.1717	−1.38001	−0.690006	+	0.723804i	$0.742392\pi$
$194$	0	0
$195$	46.4119	3.32363
$196$	0	0
$197$	2.89557	0.206301	0.103151	−	0.994666i	$-0.467108\pi$
$197$	2.89557	0.206301	0.103151	+	0.994666i	$0.467108\pi$
$198$	0	0
$199$	−26.1830	−1.85607	−0.928033	−	0.372498i	$-0.878501\pi$
$199$	−26.1830	−1.85607	−0.928033	+	0.372498i	$0.878501\pi$
$200$	0	0
$201$	11.1710	0.787944
$202$	0	0
$203$	0	0
$204$	0	0
$205$	19.0855	1.33299
$206$	0	0
$207$	2.00158	0.139120
$208$	0	0
$209$	−2.73225	−0.188994
$210$	0	0
$211$	−2.11048	−0.145291	−0.0726456	−	0.997358i	$-0.523144\pi$
$211$	−2.11048	−0.145291	−0.0726456	+	0.997358i	$0.523144\pi$
$212$	0	0
$213$	−7.02307	−0.481213
$214$	0	0
$215$	37.7406	2.57389
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−29.5233	−1.99500
$220$	0	0
$221$	−6.47214	−0.435363
$222$	0	0
$223$	18.8126	1.25979	0.629893	−	0.776682i	$-0.283099\pi$
$223$	18.8126	1.25979	0.629893	+	0.776682i	$0.283099\pi$
$224$	0	0
$225$	9.45512	0.630341
$226$	0	0
$227$	17.3737	1.15313	0.576565	−	0.817051i	$-0.304393\pi$
$227$	17.3737	1.15313	0.576565	+	0.817051i	$0.304393\pi$
$228$	0	0
$229$	2.71523	0.179428	0.0897138	−	0.995968i	$-0.471405\pi$
$229$	2.71523	0.179428	0.0897138	+	0.995968i	$0.471405\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.715233	0.0468565	0.0234282	−	0.999726i	$-0.492542\pi$
$233$	0.715233	0.0468565	0.0234282	+	0.999726i	$0.492542\pi$
$234$	0	0
$235$	−26.1487	−1.70575
$236$	0	0
$237$	−0.680990	−0.0442351
$238$	0	0
$239$	8.60745	0.556770	0.278385	−	0.960470i	$-0.410201\pi$
$239$	8.60745	0.556770	0.278385	+	0.960470i	$0.410201\pi$
$240$	0	0
$241$	−6.31241	−0.406618	−0.203309	−	0.979115i	$-0.565170\pi$
$241$	−6.31241	−0.406618	−0.203309	+	0.979115i	$0.565170\pi$
$242$	0	0
$243$	−13.7707	−0.883389
$244$	0	0
$245$	0	0
$246$	0	0
$247$	14.3063	0.910285
$248$	0	0
$249$	−29.2351	−1.85270
$250$	0	0
$251$	26.5711	1.67715	0.838577	−	0.544783i	$-0.183388\pi$
$251$	26.5711	1.67715	0.838577	+	0.544783i	$0.183388\pi$
$252$	0	0
$253$	−1.72778	−0.108625
$254$	0	0
$255$	−7.17104	−0.449068
$256$	0	0
$257$	−6.75085	−0.421107	−0.210553	−	0.977582i	$-0.567527\pi$
$257$	−6.75085	−0.421107	−0.210553	+	0.977582i	$0.567527\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.907674	−0.0561836
$262$	0	0
$263$	1.20439	0.0742657	0.0371328	−	0.999310i	$-0.488178\pi$
$263$	1.20439	0.0742657	0.0371328	+	0.999310i	$0.488178\pi$
$264$	0	0
$265$	39.9108	2.45170
$266$	0	0
$267$	31.0538	1.90046
$268$	0	0
$269$	−18.0673	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$269$	−18.0673	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$270$	0	0
$271$	−19.7464	−1.19951	−0.599754	−	0.800185i	$-0.704735\pi$
$271$	−19.7464	−1.19951	−0.599754	+	0.800185i	$0.704735\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.16174	−0.492171
$276$	0	0
$277$	−7.99744	−0.480519	−0.240260	−	0.970709i	$-0.577233\pi$
$277$	−7.99744	−0.480519	−0.240260	+	0.970709i	$0.577233\pi$
$278$	0	0
$279$	−1.38306	−0.0828016
$280$	0	0
$281$	3.07890	0.183672	0.0918359	−	0.995774i	$-0.470726\pi$
$281$	3.07890	0.183672	0.0918359	+	0.995774i	$0.470726\pi$
$282$	0	0
$283$	−28.6927	−1.70560	−0.852801	−	0.522236i	$-0.825098\pi$
$283$	−28.6927	−1.70560	−0.852801	+	0.522236i	$0.825098\pi$
$284$	0	0
$285$	15.8511	0.938940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.4388	0.729175
$292$	0	0
$293$	2.14866	0.125526	0.0627630	−	0.998028i	$-0.480009\pi$
$293$	2.14866	0.125526	0.0627630	+	0.998028i	$0.480009\pi$
$294$	0	0
$295$	2.05126	0.119429
$296$	0	0
$297$	4.08038	0.236768
$298$	0	0
$299$	9.04679	0.523190
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.7047	−0.614970
$304$	0	0
$305$	33.5301	1.91993
$306$	0	0
$307$	−16.0291	−0.914830	−0.457415	−	0.889253i	$-0.651225\pi$
$307$	−16.0291	−0.914830	−0.457415	+	0.889253i	$0.651225\pi$
$308$	0	0
$309$	11.1256	0.632913
$310$	0	0
$311$	−34.4505	−1.95351	−0.976756	−	0.214356i	$-0.931235\pi$
$311$	−34.4505	−1.95351	−0.976756	+	0.214356i	$0.931235\pi$
$312$	0	0
$313$	−32.5585	−1.84031	−0.920157	−	0.391551i	$-0.871939\pi$
$313$	−32.5585	−1.84031	−0.920157	+	0.391551i	$0.871939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.82571	0.158708	0.0793539	−	0.996847i	$-0.474714\pi$
$317$	2.82571	0.158708	0.0793539	+	0.996847i	$0.474714\pi$
$318$	0	0
$319$	0.783512	0.0438682
$320$	0	0
$321$	−19.4798	−1.08725
$322$	0	0
$323$	−2.21044	−0.122992
$324$	0	0
$325$	42.7354	2.37053
$326$	0	0
$327$	−33.8939	−1.87434
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.5822	0.911439	0.455720	−	0.890123i	$-0.349382\pi$
$331$	16.5822	0.911439	0.455720	+	0.890123i	$0.349382\pi$
$332$	0	0
$333$	−11.9000	−0.652118
$334$	0	0
$335$	18.0751	0.987549
$336$	0	0
$337$	15.7553	0.858247	0.429123	−	0.903246i	$-0.358823\pi$
$337$	15.7553	0.858247	0.429123	+	0.903246i	$0.358823\pi$
$338$	0	0
$339$	22.2120	1.20639
$340$	0	0
$341$	1.19387	0.0646516
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10.0237	0.539659
$346$	0	0
$347$	−12.6630	−0.679785	−0.339893	−	0.940464i	$-0.610391\pi$
$347$	−12.6630	−0.679785	−0.339893	+	0.940464i	$0.610391\pi$
$348$	0	0
$349$	25.1547	1.34650	0.673250	−	0.739415i	$-0.264898\pi$
$349$	25.1547	1.34650	0.673250	+	0.739415i	$0.264898\pi$
$350$	0	0
$351$	−21.3651	−1.14039
$352$	0	0
$353$	−28.5801	−1.52116	−0.760581	−	0.649243i	$-0.775086\pi$
$353$	−28.5801	−1.52116	−0.760581	+	0.649243i	$0.775086\pi$
$354$	0	0
$355$	−11.3636	−0.603115
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.4038	−1.34076	−0.670379	−	0.742018i	$-0.733869\pi$
$359$	−25.4038	−1.34076	−0.670379	+	0.742018i	$0.733869\pi$
$360$	0	0
$361$	−14.1140	−0.742840
$362$	0	0
$363$	−19.9409	−1.04663
$364$	0	0
$365$	−47.7697	−2.50038
$366$	0	0
$367$	4.87073	0.254250	0.127125	−	0.991887i	$-0.459425\pi$
$367$	4.87073	0.254250	0.127125	+	0.991887i	$0.459425\pi$
$368$	0	0
$369$	8.02317	0.417670
$370$	0	0
$371$	0	0
$372$	0	0
$373$	23.2388	1.20326	0.601629	−	0.798776i	$-0.294519\pi$
$373$	23.2388	1.20326	0.601629	+	0.798776i	$0.294519\pi$
$374$	0	0
$375$	11.4951	0.593603
$376$	0	0
$377$	−4.10252	−0.211291
$378$	0	0
$379$	15.8658	0.814971	0.407486	−	0.913212i	$-0.366406\pi$
$379$	15.8658	0.814971	0.407486	+	0.913212i	$0.366406\pi$
$380$	0	0
$381$	−37.7406	−1.93351
$382$	0	0
$383$	2.06828	0.105684	0.0528421	−	0.998603i	$-0.483172\pi$
$383$	2.06828	0.105684	0.0528421	+	0.998603i	$0.483172\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.8654	0.806482
$388$	0	0
$389$	3.51679	0.178308	0.0891542	−	0.996018i	$-0.471584\pi$
$389$	3.51679	0.178308	0.0891542	+	0.996018i	$0.471584\pi$
$390$	0	0
$391$	−1.39781	−0.0706901
$392$	0	0
$393$	−3.15313	−0.159054
$394$	0	0
$395$	−1.10187	−0.0554408
$396$	0	0
$397$	−5.47459	−0.274762	−0.137381	−	0.990518i	$-0.543868\pi$
$397$	−5.47459	−0.274762	−0.137381	+	0.990518i	$0.543868\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.76137	−0.337647	−0.168823	−	0.985646i	$-0.553997\pi$
$401$	−6.76137	−0.337647	−0.168823	+	0.985646i	$0.553997\pi$
$402$	0	0
$403$	−6.25118	−0.311393
$404$	0	0
$405$	−38.3053	−1.90340
$406$	0	0
$407$	10.2722	0.509175
$408$	0	0
$409$	14.6378	0.723793	0.361897	−	0.932218i	$-0.382129\pi$
$409$	14.6378	0.723793	0.361897	+	0.932218i	$0.382129\pi$
$410$	0	0
$411$	28.6373	1.41257
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−47.3034	−2.32203
$416$	0	0
$417$	13.2223	0.647499
$418$	0	0
$419$	−29.5736	−1.44476	−0.722382	−	0.691494i	$-0.756953\pi$
$419$	−29.5736	−1.44476	−0.722382	+	0.691494i	$0.756953\pi$
$420$	0	0
$421$	12.0223	0.585930	0.292965	−	0.956123i	$-0.405358\pi$
$421$	12.0223	0.585930	0.292965	+	0.956123i	$0.405358\pi$
$422$	0	0
$423$	−10.9924	−0.534467
$424$	0	0
$425$	−6.60299	−0.320292
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−16.8417	−0.813127
$430$	0	0
$431$	−11.1034	−0.534834	−0.267417	−	0.963581i	$-0.586170\pi$
$431$	−11.1034	−0.534834	−0.267417	+	0.963581i	$0.586170\pi$
$432$	0	0
$433$	−17.2165	−0.827372	−0.413686	−	0.910420i	$-0.635759\pi$
$433$	−17.2165	−0.827372	−0.413686	+	0.910420i	$0.635759\pi$
$434$	0	0
$435$	−4.54554	−0.217942
$436$	0	0
$437$	3.08976	0.147803
$438$	0	0
$439$	−25.1681	−1.20121	−0.600603	−	0.799548i	$-0.705073\pi$
$439$	−25.1681	−1.20121	−0.600603	+	0.799548i	$0.705073\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1531	1.48013	0.740065	−	0.672536i	$-0.234795\pi$
$443$	31.1531	1.48013	0.740065	+	0.672536i	$0.234795\pi$
$444$	0	0
$445$	50.2461	2.38189
$446$	0	0
$447$	45.6841	2.16078
$448$	0	0
$449$	10.7444	0.507057	0.253529	−	0.967328i	$-0.418409\pi$
$449$	10.7444	0.507057	0.253529	+	0.967328i	$0.418409\pi$
$450$	0	0
$451$	−6.92567	−0.326117
$452$	0	0
$453$	9.69969	0.455731
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−30.0863	−1.40738	−0.703690	−	0.710508i	$-0.748465\pi$
$457$	−30.0863	−1.40738	−0.703690	+	0.710508i	$0.748465\pi$
$458$	0	0
$459$	3.30110	0.154082
$460$	0	0
$461$	−8.29082	−0.386142	−0.193071	−	0.981185i	$-0.561845\pi$
$461$	−8.29082	−0.386142	−0.193071	+	0.981185i	$0.561845\pi$
$462$	0	0
$463$	11.2795	0.524203	0.262102	−	0.965040i	$-0.415584\pi$
$463$	11.2795	0.524203	0.262102	+	0.965040i	$0.415584\pi$
$464$	0	0
$465$	−6.92622	−0.321196
$466$	0	0
$467$	−38.2582	−1.77038	−0.885188	−	0.465233i	$-0.845971\pi$
$467$	−38.2582	−1.77038	−0.885188	+	0.465233i	$0.845971\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.54488	0.439805
$472$	0	0
$473$	−13.6951	−0.629702
$474$	0	0
$475$	14.5955	0.669687
$476$	0	0
$477$	16.7777	0.768198
$478$	0	0
$479$	33.0812	1.51152	0.755759	−	0.654850i	$-0.227268\pi$
$479$	33.0812	1.51152	0.755759	+	0.654850i	$0.227268\pi$
$480$	0	0
$481$	−53.7860	−2.45243
$482$	0	0
$483$	0	0
$484$	0	0
$485$	20.1264	0.913892
$486$	0	0
$487$	−25.3757	−1.14988	−0.574941	−	0.818195i	$-0.694975\pi$
$487$	−25.3757	−1.14988	−0.574941	+	0.818195i	$0.694975\pi$
$488$	0	0
$489$	−25.7723	−1.16547
$490$	0	0
$491$	30.4505	1.37421	0.687107	−	0.726556i	$-0.258880\pi$
$491$	30.4505	1.37421	0.687107	+	0.726556i	$0.258880\pi$
$492$	0	0
$493$	0.633874	0.0285483
$494$	0	0
$495$	−6.02912	−0.270989
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.9016	0.532790	0.266395	−	0.963864i	$-0.414167\pi$
$499$	11.9016	0.532790	0.266395	+	0.963864i	$0.414167\pi$
$500$	0	0
$501$	−2.82370	−0.126154
$502$	0	0
$503$	−8.90567	−0.397084	−0.198542	−	0.980092i	$-0.563621\pi$
$503$	−8.90567	−0.397084	−0.198542	+	0.980092i	$0.563621\pi$
$504$	0	0
$505$	−17.3206	−0.770756
$506$	0	0
$507$	60.8167	2.70096
$508$	0	0
$509$	27.1838	1.20490	0.602451	−	0.798156i	$-0.294191\pi$
$509$	27.1838	1.20490	0.602451	+	0.798156i	$0.294191\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.29687	−0.322165
$514$	0	0
$515$	18.0016	0.793245
$516$	0	0
$517$	9.48870	0.417313
$518$	0	0
$519$	−3.85887	−0.169386
$520$	0	0
$521$	−7.10981	−0.311487	−0.155743	−	0.987798i	$-0.549777\pi$
$521$	−7.10981	−0.311487	−0.155743	+	0.987798i	$0.549777\pi$
$522$	0	0
$523$	30.4581	1.33184	0.665919	−	0.746024i	$-0.268039\pi$
$523$	30.4581	1.33184	0.665919	+	0.746024i	$0.268039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.965861	0.0420735
$528$	0	0
$529$	−21.0461	−0.915049
$530$	0	0
$531$	0.862309	0.0374210
$532$	0	0
$533$	36.2633	1.57074
$534$	0	0
$535$	−31.5189	−1.36268
$536$	0	0
$537$	42.5976	1.83822
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.88794	−0.0811688	−0.0405844	−	0.999176i	$-0.512922\pi$
$541$	−1.88794	−0.0811688	−0.0405844	+	0.999176i	$0.512922\pi$
$542$	0	0
$543$	12.6947	0.544781
$544$	0	0
$545$	−54.8415	−2.34915
$546$	0	0
$547$	13.8844	0.593654	0.296827	−	0.954931i	$-0.404072\pi$
$547$	13.8844	0.593654	0.296827	+	0.954931i	$0.404072\pi$
$548$	0	0
$549$	14.0954	0.601577
$550$	0	0
$551$	−1.40114	−0.0596906
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−59.5942	−2.52963
$556$	0	0
$557$	−18.5183	−0.784644	−0.392322	−	0.919828i	$-0.628328\pi$
$557$	−18.5183	−0.784644	−0.392322	+	0.919828i	$0.628328\pi$
$558$	0	0
$559$	71.7086	3.03295
$560$	0	0
$561$	2.60219	0.109865
$562$	0	0
$563$	36.1932	1.52536	0.762681	−	0.646775i	$-0.223883\pi$
$563$	36.1932	1.52536	0.762681	+	0.646775i	$0.223883\pi$
$564$	0	0
$565$	35.9398	1.51200
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.9137	1.21213	0.606063	−	0.795417i	$-0.292748\pi$
$569$	28.9137	1.21213	0.606063	+	0.795417i	$0.292748\pi$
$570$	0	0
$571$	15.1214	0.632813	0.316406	−	0.948624i	$-0.397524\pi$
$571$	15.1214	0.632813	0.316406	+	0.948624i	$0.397524\pi$
$572$	0	0
$573$	−22.9776	−0.959904
$574$	0	0
$575$	9.22970	0.384905
$576$	0	0
$577$	18.1608	0.756042	0.378021	−	0.925797i	$-0.376605\pi$
$577$	18.1608	0.756042	0.378021	+	0.925797i	$0.376605\pi$
$578$	0	0
$579$	−40.3607	−1.67733
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.4827	−0.599810
$584$	0	0
$585$	31.5689	1.30521
$586$	0	0
$587$	−45.0538	−1.85957	−0.929784	−	0.368105i	$-0.880007\pi$
$587$	−45.0538	−1.85957	−0.929784	+	0.368105i	$0.880007\pi$
$588$	0	0
$589$	−2.13497	−0.0879701
$590$	0	0
$591$	6.09581	0.250748
$592$	0	0
$593$	−6.43139	−0.264106	−0.132053	−	0.991243i	$-0.542157\pi$
$593$	−6.43139	−0.264106	−0.132053	+	0.991243i	$0.542157\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−55.1210	−2.25595
$598$	0	0
$599$	−41.0655	−1.67789	−0.838946	−	0.544215i	$-0.816828\pi$
$599$	−41.0655	−1.67789	−0.838946	+	0.544215i	$0.816828\pi$
$600$	0	0
$601$	−39.2626	−1.60156	−0.800778	−	0.598961i	$-0.795580\pi$
$601$	−39.2626	−1.60156	−0.800778	+	0.598961i	$0.795580\pi$
$602$	0	0
$603$	7.59841	0.309431
$604$	0	0
$605$	−32.2651	−1.31176
$606$	0	0
$607$	11.9954	0.486879	0.243440	−	0.969916i	$-0.421724\pi$
$607$	11.9954	0.486879	0.243440	+	0.969916i	$0.421724\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−49.6835	−2.00998
$612$	0	0
$613$	28.0774	1.13404	0.567018	−	0.823706i	$-0.308097\pi$
$613$	28.0774	1.13404	0.567018	+	0.823706i	$0.308097\pi$
$614$	0	0
$615$	40.1792	1.62018
$616$	0	0
$617$	−6.08688	−0.245049	−0.122524	−	0.992466i	$-0.539099\pi$
$617$	−6.08688	−0.245049	−0.122524	+	0.992466i	$0.539099\pi$
$618$	0	0
$619$	−3.24659	−0.130491	−0.0652456	−	0.997869i	$-0.520783\pi$
$619$	−3.24659	−0.130491	−0.0652456	+	0.997869i	$0.520783\pi$
$620$	0	0
$621$	−4.61429	−0.185165
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14.4155	−0.576621
$626$	0	0
$627$	−5.75199	−0.229712
$628$	0	0
$629$	8.31040	0.331357
$630$	0	0
$631$	−6.53093	−0.259992	−0.129996	−	0.991515i	$-0.541496\pi$
$631$	−6.53093	−0.259992	−0.129996	+	0.991515i	$0.541496\pi$
$632$	0	0
$633$	−4.44302	−0.176594
$634$	0	0
$635$	−61.0655	−2.42331
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.77701	−0.188976
$640$	0	0
$641$	−6.98745	−0.275988	−0.137994	−	0.990433i	$-0.544065\pi$
$641$	−6.98745	−0.275988	−0.137994	+	0.990433i	$0.544065\pi$
$642$	0	0
$643$	−21.1348	−0.833475	−0.416737	−	0.909027i	$-0.636827\pi$
$643$	−21.1348	−0.833475	−0.416737	+	0.909027i	$0.636827\pi$
$644$	0	0
$645$	79.4522	3.12843
$646$	0	0
$647$	−34.2892	−1.34805	−0.674024	−	0.738709i	$-0.735436\pi$
$647$	−34.2892	−1.34805	−0.674024	+	0.738709i	$0.735436\pi$
$648$	0	0
$649$	−0.744352	−0.0292184
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.4234	0.799231	0.399615	−	0.916683i	$-0.369144\pi$
$653$	20.4234	0.799231	0.399615	+	0.916683i	$0.369144\pi$
$654$	0	0
$655$	−5.10187	−0.199346
$656$	0	0
$657$	−20.0814	−0.783450
$658$	0	0
$659$	−9.16578	−0.357048	−0.178524	−	0.983936i	$-0.557132\pi$
$659$	−9.16578	−0.357048	−0.178524	+	0.983936i	$0.557132\pi$
$660$	0	0
$661$	−33.1919	−1.29102	−0.645508	−	0.763754i	$-0.723354\pi$
$661$	−33.1919	−1.29102	−0.645508	+	0.763754i	$0.723354\pi$
$662$	0	0
$663$	−13.6253	−0.529161
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.886034	−0.0343074
$668$	0	0
$669$	39.6047	1.53121
$670$	0	0
$671$	−12.1673	−0.469712
$672$	0	0
$673$	44.8277	1.72798	0.863990	−	0.503509i	$-0.167958\pi$
$673$	44.8277	1.72798	0.863990	+	0.503509i	$0.167958\pi$
$674$	0	0
$675$	−21.7971	−0.838970
$676$	0	0
$677$	17.9276	0.689013	0.344506	−	0.938784i	$-0.388046\pi$
$677$	17.9276	0.689013	0.344506	+	0.938784i	$0.388046\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	36.5753	1.40157
$682$	0	0
$683$	−35.3742	−1.35356	−0.676778	−	0.736187i	$-0.736624\pi$
$683$	−35.3742	−1.35356	−0.676778	+	0.736187i	$0.736624\pi$
$684$	0	0
$685$	46.3361	1.77041
$686$	0	0
$687$	5.71616	0.218085
$688$	0	0
$689$	75.8322	2.88898
$690$	0	0
$691$	32.6537	1.24221	0.621103	−	0.783729i	$-0.286685\pi$
$691$	32.6537	1.24221	0.621103	+	0.783729i	$0.286685\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.3941	0.811526
$696$	0	0
$697$	−5.60299	−0.212228
$698$	0	0
$699$	1.50572	0.0569516
$700$	0	0
$701$	−12.3696	−0.467194	−0.233597	−	0.972334i	$-0.575050\pi$
$701$	−12.3696	−0.467194	−0.233597	+	0.972334i	$0.575050\pi$
$702$	0	0
$703$	−18.3696	−0.692823
$704$	0	0
$705$	−55.0486	−2.07325
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.9482	0.786727	0.393363	−	0.919383i	$-0.371311\pi$
$709$	20.9482	0.786727	0.393363	+	0.919383i	$0.371311\pi$
$710$	0	0
$711$	−0.463202	−0.0173714
$712$	0	0
$713$	−1.35009	−0.0505611
$714$	0	0
$715$	−27.2505	−1.01911
$716$	0	0
$717$	18.1206	0.676725
$718$	0	0
$719$	28.0239	1.04511	0.522557	−	0.852604i	$-0.324978\pi$
$719$	28.0239	1.04511	0.522557	+	0.852604i	$0.324978\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13.2890	−0.494223
$724$	0	0
$725$	−4.18546	−0.155444
$726$	0	0
$727$	23.9844	0.889531	0.444765	−	0.895647i	$-0.353287\pi$
$727$	23.9844	0.889531	0.444765	+	0.895647i	$0.353287\pi$
$728$	0	0
$729$	4.74583	0.175772
$730$	0	0
$731$	−11.0796	−0.409793
$732$	0	0
$733$	31.0864	1.14820	0.574102	−	0.818784i	$-0.305351\pi$
$733$	31.0864	1.14820	0.574102	+	0.818784i	$0.305351\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.55902	−0.241604
$738$	0	0
$739$	−12.0841	−0.444519	−0.222260	−	0.974988i	$-0.571343\pi$
$739$	−12.0841	−0.444519	−0.222260	+	0.974988i	$0.571343\pi$
$740$	0	0
$741$	30.1178	1.10640
$742$	0	0
$743$	52.8588	1.93920	0.969600	−	0.244695i	$-0.0786879\pi$
$743$	52.8588	1.93920	0.969600	+	0.244695i	$0.0786879\pi$
$744$	0	0
$745$	73.9184	2.70816
$746$	0	0
$747$	−19.8854	−0.727568
$748$	0	0
$749$	0	0
$750$	0	0
$751$	30.6499	1.11843	0.559216	−	0.829022i	$-0.311102\pi$
$751$	30.6499	1.11843	0.559216	+	0.829022i	$0.311102\pi$
$752$	0	0
$753$	55.9380	2.03849
$754$	0	0
$755$	15.6944	0.571179
$756$	0	0
$757$	−3.57276	−0.129854	−0.0649271	−	0.997890i	$-0.520681\pi$
$757$	−3.57276	−0.129854	−0.0649271	+	0.997890i	$0.520681\pi$
$758$	0	0
$759$	−3.63736	−0.132028
$760$	0	0
$761$	4.64537	0.168395	0.0841973	−	0.996449i	$-0.473167\pi$
$761$	4.64537	0.168395	0.0841973	+	0.996449i	$0.473167\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.87766	−0.176352
$766$	0	0
$767$	3.89748	0.140730
$768$	0	0
$769$	13.1361	0.473700	0.236850	−	0.971546i	$-0.423885\pi$
$769$	13.1361	0.473700	0.236850	+	0.971546i	$0.423885\pi$
$770$	0	0
$771$	−14.2120	−0.511833
$772$	0	0
$773$	44.0372	1.58391	0.791954	−	0.610581i	$-0.209064\pi$
$773$	44.0372	1.58391	0.791954	+	0.610581i	$0.209064\pi$
$774$	0	0
$775$	−6.37756	−0.229089
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.3850	0.443740
$780$	0	0
$781$	4.12356	0.147552
$782$	0	0
$783$	2.09248	0.0747792
$784$	0	0
$785$	15.4439	0.551218
$786$	0	0
$787$	−39.3178	−1.40153	−0.700765	−	0.713393i	$-0.747158\pi$
$787$	−39.3178	−1.40153	−0.700765	+	0.713393i	$0.747158\pi$
$788$	0	0
$789$	2.53550	0.0902661
$790$	0	0
$791$	0	0
$792$	0	0
$793$	63.7086	2.26236
$794$	0	0
$795$	84.0210	2.97992
$796$	0	0
$797$	32.5987	1.15470	0.577352	−	0.816496i	$-0.304086\pi$
$797$	32.5987	1.15470	0.577352	+	0.816496i	$0.304086\pi$
$798$	0	0
$799$	7.67652	0.271576
$800$	0	0
$801$	21.1224	0.746324
$802$	0	0
$803$	17.3344	0.611719
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−38.0356	−1.33892
$808$	0	0
$809$	−56.0461	−1.97048	−0.985239	−	0.171187i	$-0.945240\pi$
$809$	−56.0461	−1.97048	−0.985239	+	0.171187i	$0.945240\pi$
$810$	0	0
$811$	−9.39490	−0.329900	−0.164950	−	0.986302i	$-0.552746\pi$
$811$	−9.39490	−0.329900	−0.164950	+	0.986302i	$0.552746\pi$
$812$	0	0
$813$	−41.5705	−1.45794
$814$	0	0
$815$	−41.7005	−1.46071
$816$	0	0
$817$	24.4907	0.856822
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.40016	−0.258267	−0.129134	−	0.991627i	$-0.541220\pi$
$821$	−7.40016	−0.258267	−0.129134	+	0.991627i	$0.541220\pi$
$822$	0	0
$823$	−39.2675	−1.36878	−0.684390	−	0.729116i	$-0.739932\pi$
$823$	−39.2675	−1.36878	−0.684390	+	0.729116i	$0.739932\pi$
$824$	0	0
$825$	−17.1822	−0.598209
$826$	0	0
$827$	30.5052	1.06077	0.530385	−	0.847757i	$-0.322047\pi$
$827$	30.5052	1.06077	0.530385	+	0.847757i	$0.322047\pi$
$828$	0	0
$829$	46.0461	1.59925	0.799624	−	0.600501i	$-0.205032\pi$
$829$	46.0461	1.59925	0.799624	+	0.600501i	$0.205032\pi$
$830$	0	0
$831$	−16.8364	−0.584047
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.56885	−0.158111
$836$	0	0
$837$	3.18840	0.110207
$838$	0	0
$839$	−43.1903	−1.49110	−0.745548	−	0.666452i	$-0.767812\pi$
$839$	−43.1903	−1.49110	−0.745548	+	0.666452i	$0.767812\pi$
$840$	0	0
$841$	−28.5982	−0.986145
$842$	0	0
$843$	6.48176	0.223244
$844$	0	0
$845$	98.4035	3.38518
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−60.4043	−2.07307
$850$	0	0
$851$	−11.6163	−0.398203
$852$	0	0
$853$	−31.9186	−1.09287	−0.546437	−	0.837500i	$-0.684016\pi$
$853$	−31.9186	−1.09287	−0.546437	+	0.837500i	$0.684016\pi$
$854$	0	0
$855$	10.7818	0.368728
$856$	0	0
$857$	−11.6363	−0.397490	−0.198745	−	0.980051i	$-0.563686\pi$
$857$	−11.6363	−0.397490	−0.198745	+	0.980051i	$0.563686\pi$
$858$	0	0
$859$	−44.9205	−1.53267	−0.766335	−	0.642442i	$-0.777922\pi$
$859$	−44.9205	−1.53267	−0.766335	+	0.642442i	$0.777922\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.2676	0.757999	0.379000	−	0.925397i	$-0.376268\pi$
$863$	22.2676	0.757999	0.379000	+	0.925397i	$0.376268\pi$
$864$	0	0
$865$	−6.24379	−0.212295
$866$	0	0
$867$	2.10522	0.0714970
$868$	0	0
$869$	0.399840	0.0135636
$870$	0	0
$871$	34.3435	1.16368
$872$	0	0
$873$	8.46072	0.286352
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.2095	0.614890	0.307445	−	0.951566i	$-0.400526\pi$
$877$	18.2095	0.614890	0.307445	+	0.951566i	$0.400526\pi$
$878$	0	0
$879$	4.52340	0.152570
$880$	0	0
$881$	−31.7548	−1.06985	−0.534923	−	0.844901i	$-0.679659\pi$
$881$	−31.7548	−1.06985	−0.534923	+	0.844901i	$0.679659\pi$
$882$	0	0
$883$	−57.7644	−1.94393	−0.971964	−	0.235130i	$-0.924448\pi$
$883$	−57.7644	−1.94393	−0.971964	+	0.235130i	$0.924448\pi$
$884$	0	0
$885$	4.31835	0.145160
$886$	0	0
$887$	−21.9442	−0.736813	−0.368407	−	0.929665i	$-0.620097\pi$
$887$	−21.9442	−0.736813	−0.368407	+	0.929665i	$0.620097\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	13.9000	0.465669
$892$	0	0
$893$	−16.9685	−0.567828
$894$	0	0
$895$	68.9244	2.30389
$896$	0	0
$897$	19.0455	0.635910
$898$	0	0
$899$	0.612234	0.0204192
$900$	0	0
$901$	−11.7167	−0.390340
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.5404	0.682786
$906$	0	0
$907$	−10.0883	−0.334978	−0.167489	−	0.985874i	$-0.553566\pi$
$907$	−10.0883	−0.334978	−0.167489	+	0.985874i	$0.553566\pi$
$908$	0	0
$909$	−7.28123	−0.241503
$910$	0	0
$911$	−53.3155	−1.76642	−0.883210	−	0.468977i	$-0.844623\pi$
$911$	−53.3155	−1.76642	−0.883210	+	0.468977i	$0.844623\pi$
$912$	0	0
$913$	17.1652	0.568086
$914$	0	0
$915$	70.5883	2.33358
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.3247	0.472529	0.236264	−	0.971689i	$-0.424077\pi$
$919$	14.3247	0.472529	0.236264	+	0.971689i	$0.424077\pi$
$920$	0	0
$921$	−33.7448	−1.11193
$922$	0	0
$923$	−21.5912	−0.710684
$924$	0	0
$925$	−54.8734	−1.80423
$926$	0	0
$927$	7.56750	0.248549
$928$	0	0
$929$	−30.0091	−0.984567	−0.492284	−	0.870435i	$-0.663838\pi$
$929$	−30.0091	−0.984567	−0.492284	+	0.870435i	$0.663838\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−72.5259	−2.37439
$934$	0	0
$935$	4.21044	0.137696
$936$	0	0
$937$	45.1807	1.47599	0.737994	−	0.674807i	$-0.235773\pi$
$937$	45.1807	1.47599	0.737994	+	0.674807i	$0.235773\pi$
$938$	0	0
$939$	−68.5427	−2.23681
$940$	0	0
$941$	37.4598	1.22116	0.610578	−	0.791956i	$-0.290937\pi$
$941$	37.4598	1.22116	0.610578	+	0.791956i	$0.290937\pi$
$942$	0	0
$943$	7.83189	0.255041
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.33347	−0.0433318	−0.0216659	−	0.999765i	$-0.506897\pi$
$947$	−1.33347	−0.0433318	−0.0216659	+	0.999765i	$0.506897\pi$
$948$	0	0
$949$	−90.7643	−2.94633
$950$	0	0
$951$	5.94874	0.192901
$952$	0	0
$953$	4.14271	0.134196	0.0670978	−	0.997746i	$-0.478626\pi$
$953$	4.14271	0.134196	0.0670978	+	0.997746i	$0.478626\pi$
$954$	0	0
$955$	−37.1786	−1.20307
$956$	0	0
$957$	1.64946	0.0533196
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0671	−0.969907
$962$	0	0
$963$	−13.2499	−0.426973
$964$	0	0
$965$	−65.3050	−2.10224
$966$	0	0
$967$	23.9265	0.769423	0.384712	−	0.923037i	$-0.374301\pi$
$967$	23.9265	0.769423	0.384712	+	0.923037i	$0.374301\pi$
$968$	0	0
$969$	−4.65345	−0.149490
$970$	0	0
$971$	−41.7634	−1.34025	−0.670126	−	0.742248i	$-0.733760\pi$
$971$	−41.7634	−1.34025	−0.670126	+	0.742248i	$0.733760\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	89.9674	2.88126
$976$	0	0
$977$	−35.4143	−1.13300	−0.566501	−	0.824061i	$-0.691703\pi$
$977$	−35.4143	−1.13300	−0.566501	+	0.824061i	$0.691703\pi$
$978$	0	0
$979$	−18.2331	−0.582731
$980$	0	0
$981$	−23.0543	−0.736066
$982$	0	0
$983$	45.7062	1.45780	0.728901	−	0.684620i	$-0.240032\pi$
$983$	45.7062	1.45780	0.728901	+	0.684620i	$0.240032\pi$
$984$	0	0
$985$	9.86324	0.314269
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.4871	0.492462
$990$	0	0
$991$	13.3586	0.424351	0.212176	−	0.977232i	$-0.431945\pi$
$991$	13.3586	0.424351	0.212176	+	0.977232i	$0.431945\pi$
$992$	0	0
$993$	34.9091	1.10781
$994$	0	0
$995$	−89.1877	−2.82744
$996$	0	0
$997$	31.8346	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$997$	31.8346	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$998$	0	0
$999$	27.4334	0.867955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.o.1.4		4
7.6	odd	2		952.2.a.g.1.1	✓	4
21.20	even	2		8568.2.a.bj.1.4		4
28.27	even	2		1904.2.a.q.1.4		4
56.13	odd	2		7616.2.a.bj.1.4		4
56.27	even	2		7616.2.a.bp.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.g.1.1	✓	4	7.6	odd	2
1904.2.a.q.1.4		4	28.27	even	2
6664.2.a.o.1.4		4	1.1	even	1	trivial
7616.2.a.bj.1.4		4	56.13	odd	2
7616.2.a.bp.1.1		4	56.27	even	2
8568.2.a.bj.1.4		4	21.20	even	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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q+2.10522 q^{3} +3.40632 q^{5} +1.43195 q^{9} -1.23607 q^{11} +6.47214 q^{13} +7.17104 q^{15} -1.00000 q^{17} +2.21044 q^{19} +1.39781 q^{23} +6.60299 q^{25} -3.30110 q^{27} -0.633874 q^{29} -0.965861 q^{31} -2.60219 q^{33} -8.31040 q^{37} +13.6253 q^{39} +5.60299 q^{41} +11.0796 q^{43} +4.87766 q^{45} -7.67652 q^{47} -2.10522 q^{51} +11.7167 q^{53} -4.21044 q^{55} +4.65345 q^{57} +0.602193 q^{59} +9.84352 q^{61} +22.0461 q^{65} +5.30636 q^{67} +2.94269 q^{69} -3.33603 q^{71} -14.0239 q^{73} +13.9007 q^{75} -0.323477 q^{79} -11.2454 q^{81} -13.8870 q^{83} -3.40632 q^{85} -1.33444 q^{87} +14.7509 q^{89} -2.03335 q^{93} +7.52945 q^{95} +5.90855 q^{97} -1.76998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} + 7 q^{9} + 4 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{17} - 14 q^{19} + 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} - 8 q^{33} - 4 q^{37} + 4 q^{39} + 7 q^{41} + 19 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 + 7 * q^9 + 4 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^17 - 14 * q^19 + 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 - 8 * q^33 - 4 * q^37 + 4 * q^39 + 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 - 8 * q^65 + 15 * q^67 + 2 * q^69 + 2 * q^71 + 5 * q^73 - 10 * q^75 - 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 + 16 * q^89 - 20 * q^93 + 22 * q^95 + 15 * q^97 + 2 * q^99

Introduction

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