LMFDB - Embedded newform 3800.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$30.3431527681$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*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			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.347296 q^{3} -3.41147 q^{7} -2.87939 q^{9} +2.41147 q^{11} +6.29086 q^{13} -2.34730 q^{17} +1.00000 q^{19} -1.18479 q^{21} +2.49020 q^{23} -2.04189 q^{27} -8.17024 q^{29} -2.77332 q^{31} +0.837496 q^{33} -0.977711 q^{37} +2.18479 q^{39} -3.49020 q^{41} -2.75877 q^{43} +6.29086 q^{47} +4.63816 q^{49} -0.815207 q^{51} +2.38919 q^{53} +0.347296 q^{57} +3.67499 q^{59} -12.7442 q^{61} +9.82295 q^{63} +2.41147 q^{67} +0.864837 q^{69} +4.51754 q^{71} +1.81521 q^{73} -8.22668 q^{77} -5.04189 q^{79} +7.92902 q^{81} -8.07192 q^{83} -2.83750 q^{87} -2.94356 q^{89} -21.4611 q^{91} -0.963163 q^{93} -3.09833 q^{97} -6.94356 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{9} - 3 q^{11} + 3 q^{13} - 6 q^{17} + 3 q^{19} + 6 q^{23} - 3 q^{27} - 3 q^{29} - 15 q^{31} - 9 q^{37} + 3 q^{39} - 9 q^{41} + 3 q^{43} + 3 q^{47} - 3 q^{49} - 6 q^{51} + 3 q^{53} + 6 q^{59}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^9 - 3 * q^11 + 3 * q^13 - 6 * q^17 + 3 * q^19 + 6 * q^23 - 3 * q^27 - 3 * q^29 - 15 * q^31 - 9 * q^37 + 3 * q^39 - 9 * q^41 + 3 * q^43 + 3 * q^47 - 3 * q^49 - 6 * q^51 + 3 * q^53 + 6 * q^59 - 9 * q^61 + 9 * q^63 - 3 * q^67 - 21 * q^69 - 9 * q^71 + 9 * q^73 - 18 * q^77 - 12 * q^79 - 9 * q^81 + 9 * q^83 - 6 * q^87 + 6 * q^89 - 27 * q^91 + 9 * q^93 - 21 * q^97 - 6 * 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
$3$	0.347296	0.200512	0.100256	−	0.994962i	$-0.468034\pi$
$3$	0.347296	0.200512	0.100256	+	0.994962i	$0.468034\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.41147	−1.28942	−0.644708	−	0.764429i	$-0.723021\pi$
$7$	−3.41147	−1.28942	−0.644708	+	0.764429i	$0.723021\pi$
$8$	0	0
$9$	−2.87939	−0.959795
$10$	0	0
$11$	2.41147	0.727087	0.363543	−	0.931577i	$-0.381567\pi$
$11$	2.41147	0.727087	0.363543	+	0.931577i	$0.381567\pi$
$12$	0	0
$13$	6.29086	1.74477	0.872385	−	0.488819i	$-0.162572\pi$
$13$	6.29086	1.74477	0.872385	+	0.488819i	$0.162572\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.34730	−0.569303	−0.284651	−	0.958631i	$-0.591878\pi$
$17$	−2.34730	−0.569303	−0.284651	+	0.958631i	$0.591878\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.18479	−0.258543
$22$	0	0
$23$	2.49020	0.519243	0.259621	−	0.965711i	$-0.416402\pi$
$23$	2.49020	0.519243	0.259621	+	0.965711i	$0.416402\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.04189	−0.392962
$28$	0	0
$29$	−8.17024	−1.51718	−0.758588	−	0.651570i	$-0.774111\pi$
$29$	−8.17024	−1.51718	−0.758588	+	0.651570i	$0.774111\pi$
$30$	0	0
$31$	−2.77332	−0.498103	−0.249051	−	0.968490i	$-0.580119\pi$
$31$	−2.77332	−0.498103	−0.249051	+	0.968490i	$0.580119\pi$
$32$	0	0
$33$	0.837496	0.145789
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.977711	−0.160735	−0.0803674	−	0.996765i	$-0.525609\pi$
$37$	−0.977711	−0.160735	−0.0803674	+	0.996765i	$0.525609\pi$
$38$	0	0
$39$	2.18479	0.349847
$40$	0	0
$41$	−3.49020	−0.545078	−0.272539	−	0.962145i	$-0.587863\pi$
$41$	−3.49020	−0.545078	−0.272539	+	0.962145i	$0.587863\pi$
$42$	0	0
$43$	−2.75877	−0.420709	−0.210354	−	0.977625i	$-0.567462\pi$
$43$	−2.75877	−0.420709	−0.210354	+	0.977625i	$0.567462\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.29086	0.917616	0.458808	−	0.888535i	$-0.348277\pi$
$47$	6.29086	0.917616	0.458808	+	0.888535i	$0.348277\pi$
$48$	0	0
$49$	4.63816	0.662594
$50$	0	0
$51$	−0.815207	−0.114152
$52$	0	0
$53$	2.38919	0.328180	0.164090	−	0.986445i	$-0.447531\pi$
$53$	2.38919	0.328180	0.164090	+	0.986445i	$0.447531\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.347296	0.0460005
$58$	0	0
$59$	3.67499	0.478443	0.239222	−	0.970965i	$-0.423108\pi$
$59$	3.67499	0.478443	0.239222	+	0.970965i	$0.423108\pi$
$60$	0	0
$61$	−12.7442	−1.63173	−0.815865	−	0.578242i	$-0.803739\pi$
$61$	−12.7442	−1.63173	−0.815865	+	0.578242i	$0.803739\pi$
$62$	0	0
$63$	9.82295	1.23758
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.41147	0.294608	0.147304	−	0.989091i	$-0.452940\pi$
$67$	2.41147	0.294608	0.147304	+	0.989091i	$0.452940\pi$
$68$	0	0
$69$	0.864837	0.104114
$70$	0	0
$71$	4.51754	0.536133	0.268067	−	0.963400i	$-0.413615\pi$
$71$	4.51754	0.536133	0.268067	+	0.963400i	$0.413615\pi$
$72$	0	0
$73$	1.81521	0.212454	0.106227	−	0.994342i	$-0.466123\pi$
$73$	1.81521	0.212454	0.106227	+	0.994342i	$0.466123\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.22668	−0.937517
$78$	0	0
$79$	−5.04189	−0.567257	−0.283628	−	0.958934i	$-0.591538\pi$
$79$	−5.04189	−0.567257	−0.283628	+	0.958934i	$0.591538\pi$
$80$	0	0
$81$	7.92902	0.881002
$82$	0	0
$83$	−8.07192	−0.886008	−0.443004	−	0.896520i	$-0.646087\pi$
$83$	−8.07192	−0.886008	−0.443004	+	0.896520i	$0.646087\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.83750	−0.304212
$88$	0	0
$89$	−2.94356	−0.312017	−0.156009	−	0.987756i	$-0.549863\pi$
$89$	−2.94356	−0.312017	−0.156009	+	0.987756i	$0.549863\pi$
$90$	0	0
$91$	−21.4611	−2.24973
$92$	0	0
$93$	−0.963163	−0.0998754
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.09833	−0.314587	−0.157294	−	0.987552i	$-0.550277\pi$
$97$	−3.09833	−0.314587	−0.157294	+	0.987552i	$0.550277\pi$
$98$	0	0
$99$	−6.94356	−0.697854
$100$	0	0
$101$	−6.02229	−0.599240	−0.299620	−	0.954059i	$-0.596860\pi$
$101$	−6.02229	−0.599240	−0.299620	+	0.954059i	$0.596860\pi$
$102$	0	0
$103$	−7.25402	−0.714760	−0.357380	−	0.933959i	$-0.616330\pi$
$103$	−7.25402	−0.714760	−0.357380	+	0.933959i	$0.616330\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.94356	0.864607	0.432303	−	0.901728i	$-0.357701\pi$
$107$	8.94356	0.864607	0.432303	+	0.901728i	$0.357701\pi$
$108$	0	0
$109$	−9.81521	−0.940126	−0.470063	−	0.882633i	$-0.655769\pi$
$109$	−9.81521	−0.940126	−0.470063	+	0.882633i	$0.655769\pi$
$110$	0	0
$111$	−0.339556	−0.0322292
$112$	0	0
$113$	−7.57667	−0.712753	−0.356376	−	0.934342i	$-0.615988\pi$
$113$	−7.57667	−0.712753	−0.356376	+	0.934342i	$0.615988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−18.1138	−1.67462
$118$	0	0
$119$	8.00774	0.734068
$120$	0	0
$121$	−5.18479	−0.471345
$122$	0	0
$123$	−1.21213	−0.109294
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.33275	0.739412	0.369706	−	0.929149i	$-0.379458\pi$
$127$	8.33275	0.739412	0.369706	+	0.929149i	$0.379458\pi$
$128$	0	0
$129$	−0.958111	−0.0843570
$130$	0	0
$131$	−13.0915	−1.14381	−0.571906	−	0.820319i	$-0.693796\pi$
$131$	−13.0915	−1.14381	−0.571906	+	0.820319i	$0.693796\pi$
$132$	0	0
$133$	−3.41147	−0.295812
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.3327	−1.30997	−0.654983	−	0.755644i	$-0.727324\pi$
$137$	−15.3327	−1.30997	−0.654983	+	0.755644i	$0.727324\pi$
$138$	0	0
$139$	−7.77837	−0.659753	−0.329876	−	0.944024i	$-0.607007\pi$
$139$	−7.77837	−0.659753	−0.329876	+	0.944024i	$0.607007\pi$
$140$	0	0
$141$	2.18479	0.183993
$142$	0	0
$143$	15.1702	1.26860
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.61081	0.132858
$148$	0	0
$149$	−17.1361	−1.40384	−0.701922	−	0.712254i	$-0.747674\pi$
$149$	−17.1361	−1.40384	−0.701922	+	0.712254i	$0.747674\pi$
$150$	0	0
$151$	−21.4638	−1.74670	−0.873349	−	0.487094i	$-0.838057\pi$
$151$	−21.4638	−1.74670	−0.873349	+	0.487094i	$0.838057\pi$
$152$	0	0
$153$	6.75877	0.546414
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.5253	−1.55829	−0.779144	−	0.626846i	$-0.784346\pi$
$157$	−19.5253	−1.55829	−0.779144	+	0.626846i	$0.784346\pi$
$158$	0	0
$159$	0.829755	0.0658039
$160$	0	0
$161$	−8.49525	−0.669520
$162$	0	0
$163$	−3.59627	−0.281681	−0.140841	−	0.990032i	$-0.544981\pi$
$163$	−3.59627	−0.281681	−0.140841	+	0.990032i	$0.544981\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.1361	−1.01650	−0.508251	−	0.861209i	$-0.669708\pi$
$167$	−13.1361	−1.01650	−0.508251	+	0.861209i	$0.669708\pi$
$168$	0	0
$169$	26.5749	2.04422
$170$	0	0
$171$	−2.87939	−0.220192
$172$	0	0
$173$	16.4757	1.25262	0.626310	−	0.779574i	$-0.284564\pi$
$173$	16.4757	1.25262	0.626310	+	0.779574i	$0.284564\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.27631	0.0959334
$178$	0	0
$179$	−23.1438	−1.72985	−0.864926	−	0.501900i	$-0.832635\pi$
$179$	−23.1438	−1.72985	−0.864926	+	0.501900i	$0.832635\pi$
$180$	0	0
$181$	24.1634	1.79605	0.898027	−	0.439940i	$-0.145000\pi$
$181$	24.1634	1.79605	0.898027	+	0.439940i	$0.145000\pi$
$182$	0	0
$183$	−4.42602	−0.327181
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.66044	−0.413933
$188$	0	0
$189$	6.96585	0.506691
$190$	0	0
$191$	2.62361	0.189838	0.0949188	−	0.995485i	$-0.469741\pi$
$191$	2.62361	0.189838	0.0949188	+	0.995485i	$0.469741\pi$
$192$	0	0
$193$	−12.5544	−0.903684	−0.451842	−	0.892098i	$-0.649233\pi$
$193$	−12.5544	−0.903684	−0.451842	+	0.892098i	$0.649233\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.9094	−1.20475	−0.602373	−	0.798215i	$-0.705778\pi$
$197$	−16.9094	−1.20475	−0.602373	+	0.798215i	$0.705778\pi$
$198$	0	0
$199$	19.1607	1.35827	0.679135	−	0.734014i	$-0.262355\pi$
$199$	19.1607	1.35827	0.679135	+	0.734014i	$0.262355\pi$
$200$	0	0
$201$	0.837496	0.0590724
$202$	0	0
$203$	27.8726	1.95627
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.17024	−0.498366
$208$	0	0
$209$	2.41147	0.166805
$210$	0	0
$211$	−13.4534	−0.926168	−0.463084	−	0.886314i	$-0.653257\pi$
$211$	−13.4534	−0.926168	−0.463084	+	0.886314i	$0.653257\pi$
$212$	0	0
$213$	1.56893	0.107501
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.46110	0.642262
$218$	0	0
$219$	0.630415	0.0425995
$220$	0	0
$221$	−14.7665	−0.993303
$222$	0	0
$223$	28.9864	1.94107	0.970536	−	0.240956i	$-0.0774610\pi$
$223$	28.9864	1.94107	0.970536	+	0.240956i	$0.0774610\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.6091	0.704148	0.352074	−	0.935972i	$-0.385477\pi$
$227$	10.6091	0.704148	0.352074	+	0.935972i	$0.385477\pi$
$228$	0	0
$229$	−12.8375	−0.848326	−0.424163	−	0.905586i	$-0.639432\pi$
$229$	−12.8375	−0.848326	−0.424163	+	0.905586i	$0.639432\pi$
$230$	0	0
$231$	−2.85710	−0.187983
$232$	0	0
$233$	−10.4953	−0.687567	−0.343783	−	0.939049i	$-0.611709\pi$
$233$	−10.4953	−0.687567	−0.343783	+	0.939049i	$0.611709\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.75103	−0.113742
$238$	0	0
$239$	4.07873	0.263831	0.131915	−	0.991261i	$-0.457887\pi$
$239$	4.07873	0.263831	0.131915	+	0.991261i	$0.457887\pi$
$240$	0	0
$241$	−14.6382	−0.942927	−0.471463	−	0.881886i	$-0.656274\pi$
$241$	−14.6382	−0.942927	−0.471463	+	0.881886i	$0.656274\pi$
$242$	0	0
$243$	8.87939	0.569613
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.29086	0.400278
$248$	0	0
$249$	−2.80335	−0.177655
$250$	0	0
$251$	−12.0692	−0.761803	−0.380902	−	0.924616i	$-0.624386\pi$
$251$	−12.0692	−0.761803	−0.380902	+	0.924616i	$0.624386\pi$
$252$	0	0
$253$	6.00505	0.377534
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.62361	0.101278	0.0506389	−	0.998717i	$-0.483874\pi$
$257$	1.62361	0.101278	0.0506389	+	0.998717i	$0.483874\pi$
$258$	0	0
$259$	3.33544	0.207254
$260$	0	0
$261$	23.5253	1.45618
$262$	0	0
$263$	28.2618	1.74269	0.871347	−	0.490666i	$-0.163247\pi$
$263$	28.2618	1.74269	0.871347	+	0.490666i	$0.163247\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.02229	−0.0625631
$268$	0	0
$269$	−10.1138	−0.616650	−0.308325	−	0.951281i	$-0.599768\pi$
$269$	−10.1138	−0.616650	−0.308325	+	0.951281i	$0.599768\pi$
$270$	0	0
$271$	−22.9486	−1.39403	−0.697015	−	0.717057i	$-0.745489\pi$
$271$	−22.9486	−1.39403	−0.697015	+	0.717057i	$0.745489\pi$
$272$	0	0
$273$	−7.45336	−0.451098
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.1634	0.850998	0.425499	−	0.904959i	$-0.360099\pi$
$277$	14.1634	0.850998	0.425499	+	0.904959i	$0.360099\pi$
$278$	0	0
$279$	7.98545	0.478077
$280$	0	0
$281$	0.822948	0.0490930	0.0245465	−	0.999699i	$-0.492186\pi$
$281$	0.822948	0.0490930	0.0245465	+	0.999699i	$0.492186\pi$
$282$	0	0
$283$	4.83481	0.287399	0.143700	−	0.989621i	$-0.454100\pi$
$283$	4.83481	0.287399	0.143700	+	0.989621i	$0.454100\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.9067	0.702832
$288$	0	0
$289$	−11.4902	−0.675894
$290$	0	0
$291$	−1.07604	−0.0630784
$292$	0	0
$293$	−17.5699	−1.02644	−0.513221	−	0.858256i	$-0.671548\pi$
$293$	−17.5699	−1.02644	−0.513221	+	0.858256i	$0.671548\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.92396	−0.285717
$298$	0	0
$299$	15.6655	0.905959
$300$	0	0
$301$	9.41147	0.542468
$302$	0	0
$303$	−2.09152	−0.120155
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.9632	1.02521	0.512606	−	0.858624i	$-0.328680\pi$
$307$	17.9632	1.02521	0.512606	+	0.858624i	$0.328680\pi$
$308$	0	0
$309$	−2.51930	−0.143318
$310$	0	0
$311$	27.4543	1.55679	0.778395	−	0.627775i	$-0.216034\pi$
$311$	27.4543	1.55679	0.778395	+	0.627775i	$0.216034\pi$
$312$	0	0
$313$	2.90941	0.164450	0.0822249	−	0.996614i	$-0.473797\pi$
$313$	2.90941	0.164450	0.0822249	+	0.996614i	$0.473797\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.1070	1.29782	0.648909	−	0.760866i	$-0.275225\pi$
$317$	23.1070	1.29782	0.648909	+	0.760866i	$0.275225\pi$
$318$	0	0
$319$	−19.7023	−1.10312
$320$	0	0
$321$	3.10607	0.173364
$322$	0	0
$323$	−2.34730	−0.130607
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.40879	−0.188506
$328$	0	0
$329$	−21.4611	−1.18319
$330$	0	0
$331$	2.89218	0.158969	0.0794843	−	0.996836i	$-0.474673\pi$
$331$	2.89218	0.158969	0.0794843	+	0.996836i	$0.474673\pi$
$332$	0	0
$333$	2.81521	0.154272
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.3901	1.54651	0.773254	−	0.634096i	$-0.218628\pi$
$337$	28.3901	1.54651	0.773254	+	0.634096i	$0.218628\pi$
$338$	0	0
$339$	−2.63135	−0.142915
$340$	0	0
$341$	−6.68779	−0.362164
$342$	0	0
$343$	8.05737	0.435057
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.4175	−1.25712	−0.628558	−	0.777763i	$-0.716354\pi$
$347$	−23.4175	−1.25712	−0.628558	+	0.777763i	$0.716354\pi$
$348$	0	0
$349$	−5.48751	−0.293740	−0.146870	−	0.989156i	$-0.546920\pi$
$349$	−5.48751	−0.293740	−0.146870	+	0.989156i	$0.546920\pi$
$350$	0	0
$351$	−12.8452	−0.685628
$352$	0	0
$353$	22.8530	1.21634	0.608171	−	0.793806i	$-0.291904\pi$
$353$	22.8530	1.21634	0.608171	+	0.793806i	$0.291904\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.78106	0.147189
$358$	0	0
$359$	20.6604	1.09042	0.545208	−	0.838301i	$-0.316451\pi$
$359$	20.6604	1.09042	0.545208	+	0.838301i	$0.316451\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.80066	−0.0945101
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.29767	0.328736	0.164368	−	0.986399i	$-0.447442\pi$
$367$	6.29767	0.328736	0.164368	+	0.986399i	$0.447442\pi$
$368$	0	0
$369$	10.0496	0.523163
$370$	0	0
$371$	−8.15064	−0.423160
$372$	0	0
$373$	27.0401	1.40009	0.700043	−	0.714101i	$-0.253164\pi$
$373$	27.0401	1.40009	0.700043	+	0.714101i	$0.253164\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−51.3979	−2.64712
$378$	0	0
$379$	11.8375	0.608051	0.304026	−	0.952664i	$-0.401669\pi$
$379$	11.8375	0.608051	0.304026	+	0.952664i	$0.401669\pi$
$380$	0	0
$381$	2.89393	0.148261
$382$	0	0
$383$	−29.4252	−1.50356	−0.751779	−	0.659415i	$-0.770804\pi$
$383$	−29.4252	−1.50356	−0.751779	+	0.659415i	$0.770804\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.94356	0.403794
$388$	0	0
$389$	−6.75877	−0.342683	−0.171342	−	0.985212i	$-0.554810\pi$
$389$	−6.75877	−0.342683	−0.171342	+	0.985212i	$0.554810\pi$
$390$	0	0
$391$	−5.84524	−0.295606
$392$	0	0
$393$	−4.54664	−0.229347
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−20.3158	−1.01962	−0.509811	−	0.860286i	$-0.670285\pi$
$397$	−20.3158	−1.01962	−0.509811	+	0.860286i	$0.670285\pi$
$398$	0	0
$399$	−1.18479	−0.0593138
$400$	0	0
$401$	−5.57903	−0.278603	−0.139302	−	0.990250i	$-0.544486\pi$
$401$	−5.57903	−0.278603	−0.139302	+	0.990250i	$0.544486\pi$
$402$	0	0
$403$	−17.4466	−0.869075
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.35773	−0.116868
$408$	0	0
$409$	−37.3387	−1.84628	−0.923141	−	0.384462i	$-0.874387\pi$
$409$	−37.3387	−1.84628	−0.923141	+	0.384462i	$0.874387\pi$
$410$	0	0
$411$	−5.32501	−0.262663
$412$	0	0
$413$	−12.5371	−0.616912
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.70140	−0.132288
$418$	0	0
$419$	4.80840	0.234906	0.117453	−	0.993078i	$-0.462527\pi$
$419$	4.80840	0.234906	0.117453	+	0.993078i	$0.462527\pi$
$420$	0	0
$421$	27.0009	1.31594	0.657972	−	0.753042i	$-0.271414\pi$
$421$	27.0009	1.31594	0.657972	+	0.753042i	$0.271414\pi$
$422$	0	0
$423$	−18.1138	−0.880723
$424$	0	0
$425$	0	0
$426$	0	0
$427$	43.4766	2.10398
$428$	0	0
$429$	5.26857	0.254369
$430$	0	0
$431$	36.2249	1.74489	0.872447	−	0.488709i	$-0.162532\pi$
$431$	36.2249	1.74489	0.872447	+	0.488709i	$0.162532\pi$
$432$	0	0
$433$	−31.6382	−1.52043	−0.760216	−	0.649670i	$-0.774907\pi$
$433$	−31.6382	−1.52043	−0.760216	+	0.649670i	$0.774907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.49020	0.119122
$438$	0	0
$439$	9.88444	0.471758	0.235879	−	0.971782i	$-0.424203\pi$
$439$	9.88444	0.471758	0.235879	+	0.971782i	$0.424203\pi$
$440$	0	0
$441$	−13.3550	−0.635954
$442$	0	0
$443$	29.3191	1.39299	0.696497	−	0.717560i	$-0.254741\pi$
$443$	29.3191	1.39299	0.696497	+	0.717560i	$0.254741\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.95130	−0.281487
$448$	0	0
$449$	2.86247	0.135088	0.0675442	−	0.997716i	$-0.478484\pi$
$449$	2.86247	0.135088	0.0675442	+	0.997716i	$0.478484\pi$
$450$	0	0
$451$	−8.41653	−0.396319
$452$	0	0
$453$	−7.45430	−0.350233
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.6664	−1.20062	−0.600312	−	0.799766i	$-0.704957\pi$
$457$	−25.6664	−1.20062	−0.600312	+	0.799766i	$0.704957\pi$
$458$	0	0
$459$	4.79292	0.223714
$460$	0	0
$461$	5.77332	0.268890	0.134445	−	0.990921i	$-0.457075\pi$
$461$	5.77332	0.268890	0.134445	+	0.990921i	$0.457075\pi$
$462$	0	0
$463$	−31.1097	−1.44579	−0.722895	−	0.690958i	$-0.757189\pi$
$463$	−31.1097	−1.44579	−0.722895	+	0.690958i	$0.757189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.8408	−1.75106	−0.875532	−	0.483161i	$-0.839489\pi$
$467$	−37.8408	−1.75106	−0.875532	+	0.483161i	$0.839489\pi$
$468$	0	0
$469$	−8.22668	−0.379873
$470$	0	0
$471$	−6.78106	−0.312455
$472$	0	0
$473$	−6.65270	−0.305892
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.87939	−0.314985
$478$	0	0
$479$	5.10338	0.233179	0.116590	−	0.993180i	$-0.462804\pi$
$479$	5.10338	0.233179	0.116590	+	0.993180i	$0.462804\pi$
$480$	0	0
$481$	−6.15064	−0.280445
$482$	0	0
$483$	−2.95037	−0.134246
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.8503	0.854188	0.427094	−	0.904207i	$-0.359537\pi$
$487$	18.8503	0.854188	0.427094	+	0.904207i	$0.359537\pi$
$488$	0	0
$489$	−1.24897	−0.0564804
$490$	0	0
$491$	0.593578	0.0267878	0.0133939	−	0.999910i	$-0.495736\pi$
$491$	0.593578	0.0267878	0.0133939	+	0.999910i	$0.495736\pi$
$492$	0	0
$493$	19.1780	0.863733
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.4115	−0.691299
$498$	0	0
$499$	−29.7965	−1.33388	−0.666938	−	0.745113i	$-0.732395\pi$
$499$	−29.7965	−1.33388	−0.666938	+	0.745113i	$0.732395\pi$
$500$	0	0
$501$	−4.56212	−0.203820
$502$	0	0
$503$	−21.8357	−0.973608	−0.486804	−	0.873511i	$-0.661837\pi$
$503$	−21.8357	−0.973608	−0.486804	+	0.873511i	$0.661837\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.22937	0.409891
$508$	0	0
$509$	12.4861	0.553436	0.276718	−	0.960951i	$-0.410753\pi$
$509$	12.4861	0.553436	0.276718	+	0.960951i	$0.410753\pi$
$510$	0	0
$511$	−6.19253	−0.273942
$512$	0	0
$513$	−2.04189	−0.0901516
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.1702	0.667187
$518$	0	0
$519$	5.72193	0.251165
$520$	0	0
$521$	3.37908	0.148040	0.0740201	−	0.997257i	$-0.476417\pi$
$521$	3.37908	0.148040	0.0740201	+	0.997257i	$0.476417\pi$
$522$	0	0
$523$	−15.7419	−0.688343	−0.344172	−	0.938907i	$-0.611840\pi$
$523$	−15.7419	−0.688343	−0.344172	+	0.938907i	$0.611840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.50980	0.283571
$528$	0	0
$529$	−16.7989	−0.730387
$530$	0	0
$531$	−10.5817	−0.459207
$532$	0	0
$533$	−21.9564	−0.951035
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.03777	−0.346855
$538$	0	0
$539$	11.1848	0.481763
$540$	0	0
$541$	−1.31820	−0.0566739	−0.0283369	−	0.999598i	$-0.509021\pi$
$541$	−1.31820	−0.0566739	−0.0283369	+	0.999598i	$0.509021\pi$
$542$	0	0
$543$	8.39187	0.360130
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.68954	0.414295	0.207147	−	0.978310i	$-0.433582\pi$
$547$	9.68954	0.414295	0.207147	+	0.978310i	$0.433582\pi$
$548$	0	0
$549$	36.6955	1.56613
$550$	0	0
$551$	−8.17024	−0.348064
$552$	0	0
$553$	17.2003	0.731430
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.4757	0.613353	0.306677	−	0.951814i	$-0.400783\pi$
$557$	14.4757	0.613353	0.306677	+	0.951814i	$0.400783\pi$
$558$	0	0
$559$	−17.3550	−0.734040
$560$	0	0
$561$	−1.96585	−0.0829983
$562$	0	0
$563$	−25.8898	−1.09113	−0.545563	−	0.838070i	$-0.683684\pi$
$563$	−25.8898	−1.09113	−0.545563	+	0.838070i	$0.683684\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−27.0496	−1.13598
$568$	0	0
$569$	34.8384	1.46050	0.730251	−	0.683178i	$-0.239403\pi$
$569$	34.8384	1.46050	0.730251	+	0.683178i	$0.239403\pi$
$570$	0	0
$571$	24.8557	1.04018	0.520089	−	0.854112i	$-0.325899\pi$
$571$	24.8557	1.04018	0.520089	+	0.854112i	$0.325899\pi$
$572$	0	0
$573$	0.911169	0.0380646
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−36.4620	−1.51793	−0.758967	−	0.651129i	$-0.774296\pi$
$577$	−36.4620	−1.51793	−0.758967	+	0.651129i	$0.774296\pi$
$578$	0	0
$579$	−4.36009	−0.181199
$580$	0	0
$581$	27.5371	1.14243
$582$	0	0
$583$	5.76146	0.238615
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.9249	1.27641	0.638204	−	0.769868i	$-0.279678\pi$
$587$	30.9249	1.27641	0.638204	+	0.769868i	$0.279678\pi$
$588$	0	0
$589$	−2.77332	−0.114273
$590$	0	0
$591$	−5.87258	−0.241566
$592$	0	0
$593$	−30.2909	−1.24390	−0.621948	−	0.783058i	$-0.713659\pi$
$593$	−30.2909	−1.24390	−0.621948	+	0.783058i	$0.713659\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.65446	0.272349
$598$	0	0
$599$	25.3236	1.03469	0.517347	−	0.855776i	$-0.326920\pi$
$599$	25.3236	1.03469	0.517347	+	0.855776i	$0.326920\pi$
$600$	0	0
$601$	−34.2894	−1.39869	−0.699347	−	0.714782i	$-0.746526\pi$
$601$	−34.2894	−1.39869	−0.699347	+	0.714782i	$0.746526\pi$
$602$	0	0
$603$	−6.94356	−0.282764
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.2550	1.71508	0.857538	−	0.514421i	$-0.171993\pi$
$607$	42.2550	1.71508	0.857538	+	0.514421i	$0.171993\pi$
$608$	0	0
$609$	9.68004	0.392255
$610$	0	0
$611$	39.5749	1.60103
$612$	0	0
$613$	23.6759	0.956262	0.478131	−	0.878289i	$-0.341314\pi$
$613$	23.6759	0.956262	0.478131	+	0.878289i	$0.341314\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.8425	−0.879348	−0.439674	−	0.898157i	$-0.644906\pi$
$617$	−21.8425	−0.879348	−0.439674	+	0.898157i	$0.644906\pi$
$618$	0	0
$619$	38.2841	1.53877	0.769383	−	0.638788i	$-0.220564\pi$
$619$	38.2841	1.53877	0.769383	+	0.638788i	$0.220564\pi$
$620$	0	0
$621$	−5.08471	−0.204042
$622$	0	0
$623$	10.0419	0.402320
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.837496	0.0334464
$628$	0	0
$629$	2.29498	0.0915068
$630$	0	0
$631$	7.28817	0.290138	0.145069	−	0.989422i	$-0.453660\pi$
$631$	7.28817	0.290138	0.145069	+	0.989422i	$0.453660\pi$
$632$	0	0
$633$	−4.67230	−0.185707
$634$	0	0
$635$	0	0
$636$	0	0
$637$	29.1780	1.15607
$638$	0	0
$639$	−13.0077	−0.514578
$640$	0	0
$641$	14.2026	0.560970	0.280485	−	0.959858i	$-0.409505\pi$
$641$	14.2026	0.560970	0.280485	+	0.959858i	$0.409505\pi$
$642$	0	0
$643$	0.867526	0.0342119	0.0171059	−	0.999854i	$-0.494555\pi$
$643$	0.867526	0.0342119	0.0171059	+	0.999854i	$0.494555\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.5503	1.83008	0.915040	−	0.403364i	$-0.132159\pi$
$647$	46.5503	1.83008	0.915040	+	0.403364i	$0.132159\pi$
$648$	0	0
$649$	8.86215	0.347870
$650$	0	0
$651$	3.28581	0.128781
$652$	0	0
$653$	27.9050	1.09201	0.546003	−	0.837783i	$-0.316149\pi$
$653$	27.9050	1.09201	0.546003	+	0.837783i	$0.316149\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.22668	−0.203912
$658$	0	0
$659$	49.5340	1.92957	0.964784	−	0.263042i	$-0.0847257\pi$
$659$	49.5340	1.92957	0.964784	+	0.263042i	$0.0847257\pi$
$660$	0	0
$661$	40.0164	1.55646	0.778229	−	0.627980i	$-0.216118\pi$
$661$	40.0164	1.55646	0.778229	+	0.627980i	$0.216118\pi$
$662$	0	0
$663$	−5.12836	−0.199169
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.3455	−0.787783
$668$	0	0
$669$	10.0669	0.389208
$670$	0	0
$671$	−30.7324	−1.18641
$672$	0	0
$673$	−26.5449	−1.02323	−0.511615	−	0.859215i	$-0.670953\pi$
$673$	−26.5449	−1.02323	−0.511615	+	0.859215i	$0.670953\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.5398	0.866276	0.433138	−	0.901328i	$-0.357406\pi$
$677$	22.5398	0.866276	0.433138	+	0.901328i	$0.357406\pi$
$678$	0	0
$679$	10.5699	0.405634
$680$	0	0
$681$	3.68449	0.141190
$682$	0	0
$683$	44.2154	1.69186	0.845928	−	0.533297i	$-0.179047\pi$
$683$	44.2154	1.69186	0.845928	+	0.533297i	$0.179047\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.45842	−0.170099
$688$	0	0
$689$	15.0300	0.572599
$690$	0	0
$691$	31.0743	1.18212	0.591061	−	0.806627i	$-0.298709\pi$
$691$	31.0743	1.18212	0.591061	+	0.806627i	$0.298709\pi$
$692$	0	0
$693$	23.6878	0.899825
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.19253	0.310314
$698$	0	0
$699$	−3.64496	−0.137865
$700$	0	0
$701$	44.8185	1.69277	0.846386	−	0.532570i	$-0.178774\pi$
$701$	44.8185	1.69277	0.846386	+	0.532570i	$0.178774\pi$
$702$	0	0
$703$	−0.977711	−0.0368751
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.5449	0.772670
$708$	0	0
$709$	−40.1334	−1.50724	−0.753621	−	0.657309i	$-0.771695\pi$
$709$	−40.1334	−1.50724	−0.753621	+	0.657309i	$0.771695\pi$
$710$	0	0
$711$	14.5175	0.544450
$712$	0	0
$713$	−6.90612	−0.258636
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.41653	0.0529012
$718$	0	0
$719$	20.8571	0.777838	0.388919	−	0.921272i	$-0.372849\pi$
$719$	20.8571	0.777838	0.388919	+	0.921272i	$0.372849\pi$
$720$	0	0
$721$	24.7469	0.921623
$722$	0	0
$723$	−5.08378	−0.189068
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.2540	−0.417389	−0.208694	−	0.977981i	$-0.566921\pi$
$727$	−11.2540	−0.417389	−0.208694	+	0.977981i	$0.566921\pi$
$728$	0	0
$729$	−20.7033	−0.766788
$730$	0	0
$731$	6.47565	0.239511
$732$	0	0
$733$	−11.3952	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$733$	−11.3952	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.81521	0.214206
$738$	0	0
$739$	−45.7015	−1.68116	−0.840579	−	0.541690i	$-0.817785\pi$
$739$	−45.7015	−1.68116	−0.840579	+	0.541690i	$0.817785\pi$
$740$	0	0
$741$	2.18479	0.0802604
$742$	0	0
$743$	3.82470	0.140315	0.0701574	−	0.997536i	$-0.477650\pi$
$743$	3.82470	0.140315	0.0701574	+	0.997536i	$0.477650\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	23.2422	0.850386
$748$	0	0
$749$	−30.5107	−1.11484
$750$	0	0
$751$	26.5550	0.969005	0.484503	−	0.874790i	$-0.339001\pi$
$751$	26.5550	0.969005	0.484503	+	0.874790i	$0.339001\pi$
$752$	0	0
$753$	−4.19160	−0.152750
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.64765	0.168922	0.0844609	−	0.996427i	$-0.473083\pi$
$757$	4.64765	0.168922	0.0844609	+	0.996427i	$0.473083\pi$
$758$	0	0
$759$	2.08553	0.0757000
$760$	0	0
$761$	−26.0327	−0.943685	−0.471843	−	0.881683i	$-0.656411\pi$
$761$	−26.0327	−0.943685	−0.471843	+	0.881683i	$0.656411\pi$
$762$	0	0
$763$	33.4843	1.21221
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.1189	0.834774
$768$	0	0
$769$	3.10876	0.112105	0.0560523	−	0.998428i	$-0.482149\pi$
$769$	3.10876	0.112105	0.0560523	+	0.998428i	$0.482149\pi$
$770$	0	0
$771$	0.563873	0.0203074
$772$	0	0
$773$	−19.4516	−0.699626	−0.349813	−	0.936820i	$-0.613755\pi$
$773$	−19.4516	−0.699626	−0.349813	+	0.936820i	$0.613755\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.15839	0.0415568
$778$	0	0
$779$	−3.49020	−0.125049
$780$	0	0
$781$	10.8939	0.389816
$782$	0	0
$783$	16.6827	0.596192
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14.0496	−0.500815	−0.250408	−	0.968141i	$-0.580565\pi$
$787$	−14.0496	−0.500815	−0.250408	+	0.968141i	$0.580565\pi$
$788$	0	0
$789$	9.81521	0.349431
$790$	0	0
$791$	25.8476	0.919035
$792$	0	0
$793$	−80.1721	−2.84700
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.57810	0.0913209	0.0456604	−	0.998957i	$-0.485461\pi$
$797$	2.57810	0.0913209	0.0456604	+	0.998957i	$0.485461\pi$
$798$	0	0
$799$	−14.7665	−0.522402
$800$	0	0
$801$	8.47565	0.299472
$802$	0	0
$803$	4.37733	0.154472
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.51249	−0.123645
$808$	0	0
$809$	19.8220	0.696905	0.348452	−	0.937326i	$-0.386707\pi$
$809$	19.8220	0.696905	0.348452	+	0.937326i	$0.386707\pi$
$810$	0	0
$811$	22.6563	0.795571	0.397786	−	0.917478i	$-0.369779\pi$
$811$	22.6563	0.795571	0.397786	+	0.917478i	$0.369779\pi$
$812$	0	0
$813$	−7.96997	−0.279519
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.75877	−0.0965172
$818$	0	0
$819$	61.7948	2.15928
$820$	0	0
$821$	−23.0570	−0.804696	−0.402348	−	0.915487i	$-0.631806\pi$
$821$	−23.0570	−0.804696	−0.402348	+	0.915487i	$0.631806\pi$
$822$	0	0
$823$	−44.6563	−1.55662	−0.778311	−	0.627879i	$-0.783923\pi$
$823$	−44.6563	−1.55662	−0.778311	+	0.627879i	$0.783923\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.4306	−1.82319	−0.911595	−	0.411090i	$-0.865148\pi$
$827$	−52.4306	−1.82319	−0.911595	+	0.411090i	$0.865148\pi$
$828$	0	0
$829$	−40.2412	−1.39764	−0.698818	−	0.715300i	$-0.746290\pi$
$829$	−40.2412	−1.39764	−0.698818	+	0.715300i	$0.746290\pi$
$830$	0	0
$831$	4.91891	0.170635
$832$	0	0
$833$	−10.8871	−0.377217
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.66281	0.195735
$838$	0	0
$839$	−52.5185	−1.81314	−0.906570	−	0.422056i	$-0.861308\pi$
$839$	−52.5185	−1.81314	−0.906570	+	0.422056i	$0.861308\pi$
$840$	0	0
$841$	37.7529	1.30182
$842$	0	0
$843$	0.285807	0.00984371
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.6878	0.607760
$848$	0	0
$849$	1.67911	0.0576269
$850$	0	0
$851$	−2.43470	−0.0834603
$852$	0	0
$853$	20.2094	0.691958	0.345979	−	0.938242i	$-0.387547\pi$
$853$	20.2094	0.691958	0.345979	+	0.938242i	$0.387547\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.9522	0.510759	0.255379	−	0.966841i	$-0.417800\pi$
$857$	14.9522	0.510759	0.255379	+	0.966841i	$0.417800\pi$
$858$	0	0
$859$	−34.3286	−1.17128	−0.585639	−	0.810572i	$-0.699156\pi$
$859$	−34.3286	−1.17128	−0.585639	+	0.810572i	$0.699156\pi$
$860$	0	0
$861$	4.13516	0.140926
$862$	0	0
$863$	−27.0392	−0.920425	−0.460213	−	0.887809i	$-0.652227\pi$
$863$	−27.0392	−0.920425	−0.460213	+	0.887809i	$0.652227\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.99050	−0.135525
$868$	0	0
$869$	−12.1584	−0.412445
$870$	0	0
$871$	15.1702	0.514024
$872$	0	0
$873$	8.92127	0.301939
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.2216	0.581533	0.290767	−	0.956794i	$-0.406090\pi$
$877$	17.2216	0.581533	0.290767	+	0.956794i	$0.406090\pi$
$878$	0	0
$879$	−6.10195	−0.205814
$880$	0	0
$881$	51.2312	1.72602	0.863012	−	0.505183i	$-0.168575\pi$
$881$	51.2312	1.72602	0.863012	+	0.505183i	$0.168575\pi$
$882$	0	0
$883$	−27.8939	−0.938706	−0.469353	−	0.883011i	$-0.655513\pi$
$883$	−27.8939	−0.938706	−0.469353	+	0.883011i	$0.655513\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.5749	−1.09376	−0.546879	−	0.837211i	$-0.684184\pi$
$887$	−32.5749	−1.09376	−0.546879	+	0.837211i	$0.684184\pi$
$888$	0	0
$889$	−28.4270	−0.953409
$890$	0	0
$891$	19.1206	0.640565
$892$	0	0
$893$	6.29086	0.210516
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.44057	0.181655
$898$	0	0
$899$	22.6587	0.755710
$900$	0	0
$901$	−5.60813	−0.186834
$902$	0	0
$903$	3.26857	0.108771
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.6673	0.785858	0.392929	−	0.919569i	$-0.371462\pi$
$907$	23.6673	0.785858	0.392929	+	0.919569i	$0.371462\pi$
$908$	0	0
$909$	17.3405	0.575148
$910$	0	0
$911$	11.4233	0.378472	0.189236	−	0.981932i	$-0.439399\pi$
$911$	11.4233	0.378472	0.189236	+	0.981932i	$0.439399\pi$
$912$	0	0
$913$	−19.4652	−0.644205
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44.6614	1.47485
$918$	0	0
$919$	−34.7401	−1.14597	−0.572985	−	0.819566i	$-0.694215\pi$
$919$	−34.7401	−1.14597	−0.572985	+	0.819566i	$0.694215\pi$
$920$	0	0
$921$	6.23854	0.205567
$922$	0	0
$923$	28.4192	0.935430
$924$	0	0
$925$	0	0
$926$	0	0
$927$	20.8871	0.686023
$928$	0	0
$929$	50.1848	1.64651	0.823255	−	0.567672i	$-0.192156\pi$
$929$	50.1848	1.64651	0.823255	+	0.567672i	$0.192156\pi$
$930$	0	0
$931$	4.63816	0.152009
$932$	0	0
$933$	9.53478	0.312155
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.48927	0.146658	0.0733290	−	0.997308i	$-0.476638\pi$
$937$	4.48927	0.146658	0.0733290	+	0.997308i	$0.476638\pi$
$938$	0	0
$939$	1.01043	0.0329741
$940$	0	0
$941$	−7.70233	−0.251089	−0.125544	−	0.992088i	$-0.540068\pi$
$941$	−7.70233	−0.251089	−0.125544	+	0.992088i	$0.540068\pi$
$942$	0	0
$943$	−8.69129	−0.283028
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.4415	−0.436790	−0.218395	−	0.975860i	$-0.570082\pi$
$947$	−13.4415	−0.436790	−0.218395	+	0.975860i	$0.570082\pi$
$948$	0	0
$949$	11.4192	0.370683
$950$	0	0
$951$	8.02498	0.260228
$952$	0	0
$953$	−23.1943	−0.751337	−0.375668	−	0.926754i	$-0.622587\pi$
$953$	−23.1943	−0.751337	−0.375668	+	0.926754i	$0.622587\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.84255	−0.221188
$958$	0	0
$959$	52.3073	1.68909
$960$	0	0
$961$	−23.3087	−0.751894
$962$	0	0
$963$	−25.7520	−0.829845
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−22.1429	−0.712068	−0.356034	−	0.934473i	$-0.615871\pi$
$967$	−22.1429	−0.712068	−0.356034	+	0.934473i	$0.615871\pi$
$968$	0	0
$969$	−0.815207	−0.0261882
$970$	0	0
$971$	26.9804	0.865842	0.432921	−	0.901432i	$-0.357483\pi$
$971$	26.9804	0.865842	0.432921	+	0.901432i	$0.357483\pi$
$972$	0	0
$973$	26.5357	0.850696
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.64496	−0.0846199	−0.0423099	−	0.999105i	$-0.513472\pi$
$977$	−2.64496	−0.0846199	−0.0423099	+	0.999105i	$0.513472\pi$
$978$	0	0
$979$	−7.09833	−0.226863
$980$	0	0
$981$	28.2618	0.902329
$982$	0	0
$983$	22.4492	0.716020	0.358010	−	0.933718i	$-0.383455\pi$
$983$	22.4492	0.716020	0.358010	+	0.933718i	$0.383455\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.45336	−0.237243
$988$	0	0
$989$	−6.86989	−0.218450
$990$	0	0
$991$	−26.4570	−0.840434	−0.420217	−	0.907424i	$-0.638046\pi$
$991$	−26.4570	−0.840434	−0.420217	+	0.907424i	$0.638046\pi$
$992$	0	0
$993$	1.00444	0.0318750
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.17881	0.0690035	0.0345017	−	0.999405i	$-0.489016\pi$
$997$	2.17881	0.0690035	0.0345017	+	0.999405i	$0.489016\pi$
$998$	0	0
$999$	1.99638	0.0631626

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.t.1.2	yes	3
4.3	odd	2		7600.2.a.bs.1.2		3
5.2	odd	4		3800.2.d.o.3649.3		6
5.3	odd	4		3800.2.d.o.3649.4		6
5.4	even	2		3800.2.a.s.1.2	✓	3
20.19	odd	2		7600.2.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.2	✓	3	5.4	even	2
3800.2.a.t.1.2	yes	3	1.1	even	1	trivial
3800.2.d.o.3649.3		6	5.2	odd	4
3800.2.d.o.3649.4		6	5.3	odd	4
7600.2.a.br.1.2		3	20.19	odd	2
7600.2.a.bs.1.2		3	4.3	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.347296 q^{3} -3.41147 q^{7} -2.87939 q^{9} +2.41147 q^{11} +6.29086 q^{13} -2.34730 q^{17} +1.00000 q^{19} -1.18479 q^{21} +2.49020 q^{23} -2.04189 q^{27} -8.17024 q^{29} -2.77332 q^{31} +0.837496 q^{33} -0.977711 q^{37} +2.18479 q^{39} -3.49020 q^{41} -2.75877 q^{43} +6.29086 q^{47} +4.63816 q^{49} -0.815207 q^{51} +2.38919 q^{53} +0.347296 q^{57} +3.67499 q^{59} -12.7442 q^{61} +9.82295 q^{63} +2.41147 q^{67} +0.864837 q^{69} +4.51754 q^{71} +1.81521 q^{73} -8.22668 q^{77} -5.04189 q^{79} +7.92902 q^{81} -8.07192 q^{83} -2.83750 q^{87} -2.94356 q^{89} -21.4611 q^{91} -0.963163 q^{93} -3.09833 q^{97} -6.94356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{9} - 3 q^{11} + 3 q^{13} - 6 q^{17} + 3 q^{19} + 6 q^{23} - 3 q^{27} - 3 q^{29} - 15 q^{31} - 9 q^{37} + 3 q^{39} - 9 q^{41} + 3 q^{43} + 3 q^{47} - 3 q^{49} - 6 q^{51} + 3 q^{53} + 6 q^{59}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^9 - 3 * q^11 + 3 * q^13 - 6 * q^17 + 3 * q^19 + 6 * q^23 - 3 * q^27 - 3 * q^29 - 15 * q^31 - 9 * q^37 + 3 * q^39 - 9 * q^41 + 3 * q^43 + 3 * q^47 - 3 * q^49 - 6 * q^51 + 3 * q^53 + 6 * q^59 - 9 * q^61 + 9 * q^63 - 3 * q^67 - 21 * q^69 - 9 * q^71 + 9 * q^73 - 18 * q^77 - 12 * q^79 - 9 * q^81 + 9 * q^83 - 6 * q^87 + 6 * q^89 - 27 * q^91 + 9 * q^93 - 21 * q^97 - 6 * q^99

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