LMFDB - Embedded newform 8020.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65793 q^{3} +1.00000 q^{5} +4.10503 q^{7} -0.251260 q^{9} +O(q^{10})$	$q-1.65793 q^{3} +1.00000 q^{5} +4.10503 q^{7} -0.251260 q^{9} +5.07698 q^{11} -3.93111 q^{13} -1.65793 q^{15} -1.21438 q^{17} +3.27467 q^{19} -6.80586 q^{21} +6.17753 q^{23} +1.00000 q^{25} +5.39037 q^{27} +2.73528 q^{29} -6.62915 q^{31} -8.41730 q^{33} +4.10503 q^{35} +0.852473 q^{37} +6.51751 q^{39} -4.07686 q^{41} -0.400782 q^{43} -0.251260 q^{45} +8.42691 q^{47} +9.85125 q^{49} +2.01336 q^{51} -12.9098 q^{53} +5.07698 q^{55} -5.42919 q^{57} +2.47201 q^{59} +3.88479 q^{61} -1.03143 q^{63} -3.93111 q^{65} +10.2444 q^{67} -10.2419 q^{69} +8.58441 q^{71} +5.83210 q^{73} -1.65793 q^{75} +20.8412 q^{77} +9.07348 q^{79} -8.18309 q^{81} +5.00895 q^{83} -1.21438 q^{85} -4.53491 q^{87} -5.51867 q^{89} -16.1373 q^{91} +10.9907 q^{93} +3.27467 q^{95} +3.13851 q^{97} -1.27564 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.65793	−0.957208	−0.478604	−	0.878031i	$-0.658857\pi$
$3$	−1.65793	−0.957208	−0.478604	+	0.878031i	$0.658857\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.10503	1.55155	0.775777	−	0.631007i	$-0.217358\pi$
$7$	4.10503	1.55155	0.775777	+	0.631007i	$0.217358\pi$
$8$	0	0
$9$	−0.251260	−0.0837534
$10$	0	0
$11$	5.07698	1.53077	0.765384	−	0.643574i	$-0.222549\pi$
$11$	5.07698	1.53077	0.765384	+	0.643574i	$0.222549\pi$
$12$	0	0
$13$	−3.93111	−1.09029	−0.545147	−	0.838341i	$-0.683526\pi$
$13$	−3.93111	−1.09029	−0.545147	+	0.838341i	$0.683526\pi$
$14$	0	0
$15$	−1.65793	−0.428076
$16$	0	0
$17$	−1.21438	−0.294530	−0.147265	−	0.989097i	$-0.547047\pi$
$17$	−1.21438	−0.294530	−0.147265	+	0.989097i	$0.547047\pi$
$18$	0	0
$19$	3.27467	0.751262	0.375631	−	0.926769i	$-0.377426\pi$
$19$	3.27467	0.751262	0.375631	+	0.926769i	$0.377426\pi$
$20$	0	0
$21$	−6.80586	−1.48516
$22$	0	0
$23$	6.17753	1.28810	0.644052	−	0.764982i	$-0.277252\pi$
$23$	6.17753	1.28810	0.644052	+	0.764982i	$0.277252\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.39037	1.03738
$28$	0	0
$29$	2.73528	0.507929	0.253964	−	0.967214i	$-0.418265\pi$
$29$	2.73528	0.507929	0.253964	+	0.967214i	$0.418265\pi$
$30$	0	0
$31$	−6.62915	−1.19063	−0.595315	−	0.803492i	$-0.702973\pi$
$31$	−6.62915	−1.19063	−0.595315	+	0.803492i	$0.702973\pi$
$32$	0	0
$33$	−8.41730	−1.46526
$34$	0	0
$35$	4.10503	0.693876
$36$	0	0
$37$	0.852473	0.140146	0.0700728	−	0.997542i	$-0.477677\pi$
$37$	0.852473	0.140146	0.0700728	+	0.997542i	$0.477677\pi$
$38$	0	0
$39$	6.51751	1.04364
$40$	0	0
$41$	−4.07686	−0.636698	−0.318349	−	0.947974i	$-0.603128\pi$
$41$	−4.07686	−0.636698	−0.318349	+	0.947974i	$0.603128\pi$
$42$	0	0
$43$	−0.400782	−0.0611187	−0.0305593	−	0.999533i	$-0.509729\pi$
$43$	−0.400782	−0.0611187	−0.0305593	+	0.999533i	$0.509729\pi$
$44$	0	0
$45$	−0.251260	−0.0374556
$46$	0	0
$47$	8.42691	1.22919	0.614596	−	0.788842i	$-0.289319\pi$
$47$	8.42691	1.22919	0.614596	+	0.788842i	$0.289319\pi$
$48$	0	0
$49$	9.85125	1.40732
$50$	0	0
$51$	2.01336	0.281927
$52$	0	0
$53$	−12.9098	−1.77330	−0.886651	−	0.462439i	$-0.846974\pi$
$53$	−12.9098	−1.77330	−0.886651	+	0.462439i	$0.846974\pi$
$54$	0	0
$55$	5.07698	0.684580
$56$	0	0
$57$	−5.42919	−0.719113
$58$	0	0
$59$	2.47201	0.321828	0.160914	−	0.986968i	$-0.448556\pi$
$59$	2.47201	0.321828	0.160914	+	0.986968i	$0.448556\pi$
$60$	0	0
$61$	3.88479	0.497397	0.248698	−	0.968581i	$-0.419997\pi$
$61$	3.88479	0.497397	0.248698	+	0.968581i	$0.419997\pi$
$62$	0	0
$63$	−1.03143	−0.129948
$64$	0	0
$65$	−3.93111	−0.487594
$66$	0	0
$67$	10.2444	1.25156	0.625778	−	0.780001i	$-0.284782\pi$
$67$	10.2444	1.25156	0.625778	+	0.780001i	$0.284782\pi$
$68$	0	0
$69$	−10.2419	−1.23298
$70$	0	0
$71$	8.58441	1.01878	0.509391	−	0.860535i	$-0.329871\pi$
$71$	8.58441	1.01878	0.509391	+	0.860535i	$0.329871\pi$
$72$	0	0
$73$	5.83210	0.682596	0.341298	−	0.939955i	$-0.389133\pi$
$73$	5.83210	0.682596	0.341298	+	0.939955i	$0.389133\pi$
$74$	0	0
$75$	−1.65793	−0.191442
$76$	0	0
$77$	20.8412	2.37507
$78$	0	0
$79$	9.07348	1.02085	0.510423	−	0.859923i	$-0.329489\pi$
$79$	9.07348	1.02085	0.510423	+	0.859923i	$0.329489\pi$
$80$	0	0
$81$	−8.18309	−0.909232
$82$	0	0
$83$	5.00895	0.549804	0.274902	−	0.961472i	$-0.411355\pi$
$83$	5.00895	0.549804	0.274902	+	0.961472i	$0.411355\pi$
$84$	0	0
$85$	−1.21438	−0.131718
$86$	0	0
$87$	−4.53491	−0.486194
$88$	0	0
$89$	−5.51867	−0.584978	−0.292489	−	0.956269i	$-0.594483\pi$
$89$	−5.51867	−0.584978	−0.292489	+	0.956269i	$0.594483\pi$
$90$	0	0
$91$	−16.1373	−1.69165
$92$	0	0
$93$	10.9907	1.13968
$94$	0	0
$95$	3.27467	0.335974
$96$	0	0
$97$	3.13851	0.318667	0.159333	−	0.987225i	$-0.449065\pi$
$97$	3.13851	0.318667	0.159333	+	0.987225i	$0.449065\pi$
$98$	0	0
$99$	−1.27564	−0.128207
$100$	0	0
$101$	−4.56320	−0.454056	−0.227028	−	0.973888i	$-0.572901\pi$
$101$	−4.56320	−0.454056	−0.227028	+	0.973888i	$0.572901\pi$
$102$	0	0
$103$	−2.74592	−0.270563	−0.135282	−	0.990807i	$-0.543194\pi$
$103$	−2.74592	−0.270563	−0.135282	+	0.990807i	$0.543194\pi$
$104$	0	0
$105$	−6.80586	−0.664184
$106$	0	0
$107$	−2.61994	−0.253279	−0.126639	−	0.991949i	$-0.540419\pi$
$107$	−2.61994	−0.253279	−0.126639	+	0.991949i	$0.540419\pi$
$108$	0	0
$109$	−7.63700	−0.731492	−0.365746	−	0.930715i	$-0.619186\pi$
$109$	−7.63700	−0.731492	−0.365746	+	0.930715i	$0.619186\pi$
$110$	0	0
$111$	−1.41334	−0.134149
$112$	0	0
$113$	−0.450744	−0.0424025	−0.0212012	−	0.999775i	$-0.506749\pi$
$113$	−0.450744	−0.0424025	−0.0212012	+	0.999775i	$0.506749\pi$
$114$	0	0
$115$	6.17753	0.576057
$116$	0	0
$117$	0.987731	0.0913158
$118$	0	0
$119$	−4.98506	−0.456980
$120$	0	0
$121$	14.7758	1.34325
$122$	0	0
$123$	6.75915	0.609452
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.07374	0.450222	0.225111	−	0.974333i	$-0.427726\pi$
$127$	5.07374	0.450222	0.225111	+	0.974333i	$0.427726\pi$
$128$	0	0
$129$	0.664470	0.0585033
$130$	0	0
$131$	−10.1107	−0.883375	−0.441687	−	0.897169i	$-0.645620\pi$
$131$	−10.1107	−0.883375	−0.441687	+	0.897169i	$0.645620\pi$
$132$	0	0
$133$	13.4426	1.16562
$134$	0	0
$135$	5.39037	0.463929
$136$	0	0
$137$	0.201233	0.0171925	0.00859626	−	0.999963i	$-0.497264\pi$
$137$	0.201233	0.0171925	0.00859626	+	0.999963i	$0.497264\pi$
$138$	0	0
$139$	11.8102	1.00173	0.500865	−	0.865526i	$-0.333015\pi$
$139$	11.8102	1.00173	0.500865	+	0.865526i	$0.333015\pi$
$140$	0	0
$141$	−13.9713	−1.17659
$142$	0	0
$143$	−19.9582	−1.66899
$144$	0	0
$145$	2.73528	0.227153
$146$	0	0
$147$	−16.3327	−1.34710
$148$	0	0
$149$	−7.66326	−0.627799	−0.313899	−	0.949456i	$-0.601636\pi$
$149$	−7.66326	−0.627799	−0.313899	+	0.949456i	$0.601636\pi$
$150$	0	0
$151$	−4.13971	−0.336885	−0.168442	−	0.985712i	$-0.553874\pi$
$151$	−4.13971	−0.336885	−0.168442	+	0.985712i	$0.553874\pi$
$152$	0	0
$153$	0.305125	0.0246679
$154$	0	0
$155$	−6.62915	−0.532466
$156$	0	0
$157$	1.66979	0.133263	0.0666317	−	0.997778i	$-0.478775\pi$
$157$	1.66979	0.133263	0.0666317	+	0.997778i	$0.478775\pi$
$158$	0	0
$159$	21.4036	1.69742
$160$	0	0
$161$	25.3589	1.99856
$162$	0	0
$163$	12.3662	0.968597	0.484299	−	0.874903i	$-0.339075\pi$
$163$	12.3662	0.968597	0.484299	+	0.874903i	$0.339075\pi$
$164$	0	0
$165$	−8.41730	−0.655286
$166$	0	0
$167$	−11.5997	−0.897611	−0.448805	−	0.893630i	$-0.648150\pi$
$167$	−11.5997	−0.897611	−0.448805	+	0.893630i	$0.648150\pi$
$168$	0	0
$169$	2.45363	0.188741
$170$	0	0
$171$	−0.822795	−0.0629207
$172$	0	0
$173$	−0.0540590	−0.00411003	−0.00205501	−	0.999998i	$-0.500654\pi$
$173$	−0.0540590	−0.00411003	−0.00205501	+	0.999998i	$0.500654\pi$
$174$	0	0
$175$	4.10503	0.310311
$176$	0	0
$177$	−4.09843	−0.308057
$178$	0	0
$179$	−22.3328	−1.66923	−0.834615	−	0.550834i	$-0.814310\pi$
$179$	−22.3328	−1.66923	−0.834615	+	0.550834i	$0.814310\pi$
$180$	0	0
$181$	−17.8853	−1.32940	−0.664701	−	0.747110i	$-0.731441\pi$
$181$	−17.8853	−1.32940	−0.664701	+	0.747110i	$0.731441\pi$
$182$	0	0
$183$	−6.44073	−0.476112
$184$	0	0
$185$	0.852473	0.0626751
$186$	0	0
$187$	−6.16538	−0.450857
$188$	0	0
$189$	22.1276	1.60955
$190$	0	0
$191$	1.13422	0.0820691	0.0410345	−	0.999158i	$-0.486935\pi$
$191$	1.13422	0.0820691	0.0410345	+	0.999158i	$0.486935\pi$
$192$	0	0
$193$	11.6813	0.840840	0.420420	−	0.907330i	$-0.361883\pi$
$193$	11.6813	0.840840	0.420420	+	0.907330i	$0.361883\pi$
$194$	0	0
$195$	6.51751	0.466729
$196$	0	0
$197$	16.1556	1.15104	0.575519	−	0.817788i	$-0.304800\pi$
$197$	16.1556	1.15104	0.575519	+	0.817788i	$0.304800\pi$
$198$	0	0
$199$	8.06755	0.571893	0.285947	−	0.958246i	$-0.407692\pi$
$199$	8.06755	0.571893	0.285947	+	0.958246i	$0.407692\pi$
$200$	0	0
$201$	−16.9846	−1.19800
$202$	0	0
$203$	11.2284	0.788079
$204$	0	0
$205$	−4.07686	−0.284740
$206$	0	0
$207$	−1.55217	−0.107883
$208$	0	0
$209$	16.6255	1.15001
$210$	0	0
$211$	22.9197	1.57786	0.788929	−	0.614485i	$-0.210636\pi$
$211$	22.9197	1.57786	0.788929	+	0.614485i	$0.210636\pi$
$212$	0	0
$213$	−14.2324	−0.975186
$214$	0	0
$215$	−0.400782	−0.0273331
$216$	0	0
$217$	−27.2128	−1.84733
$218$	0	0
$219$	−9.66923	−0.653386
$220$	0	0
$221$	4.77386	0.321124
$222$	0	0
$223$	−7.78658	−0.521428	−0.260714	−	0.965416i	$-0.583958\pi$
$223$	−7.78658	−0.521428	−0.260714	+	0.965416i	$0.583958\pi$
$224$	0	0
$225$	−0.251260	−0.0167507
$226$	0	0
$227$	−13.8372	−0.918404	−0.459202	−	0.888332i	$-0.651865\pi$
$227$	−13.8372	−0.918404	−0.459202	+	0.888332i	$0.651865\pi$
$228$	0	0
$229$	2.56182	0.169290	0.0846451	−	0.996411i	$-0.473024\pi$
$229$	2.56182	0.169290	0.0846451	+	0.996411i	$0.473024\pi$
$230$	0	0
$231$	−34.5532	−2.27344
$232$	0	0
$233$	2.36331	0.154826	0.0774129	−	0.996999i	$-0.475334\pi$
$233$	2.36331	0.154826	0.0774129	+	0.996999i	$0.475334\pi$
$234$	0	0
$235$	8.42691	0.549711
$236$	0	0
$237$	−15.0432	−0.977162
$238$	0	0
$239$	11.6192	0.751586	0.375793	−	0.926704i	$-0.377370\pi$
$239$	11.6192	0.751586	0.375793	+	0.926704i	$0.377370\pi$
$240$	0	0
$241$	−27.7091	−1.78490	−0.892450	−	0.451147i	$-0.851015\pi$
$241$	−27.7091	−1.78490	−0.892450	+	0.451147i	$0.851015\pi$
$242$	0	0
$243$	−2.60410	−0.167053
$244$	0	0
$245$	9.85125	0.629373
$246$	0	0
$247$	−12.8731	−0.819096
$248$	0	0
$249$	−8.30451	−0.526277
$250$	0	0
$251$	−4.24517	−0.267952	−0.133976	−	0.990985i	$-0.542775\pi$
$251$	−4.24517	−0.267952	−0.133976	+	0.990985i	$0.542775\pi$
$252$	0	0
$253$	31.3632	1.97179
$254$	0	0
$255$	2.01336	0.126081
$256$	0	0
$257$	4.16708	0.259935	0.129968	−	0.991518i	$-0.458513\pi$
$257$	4.16708	0.259935	0.129968	+	0.991518i	$0.458513\pi$
$258$	0	0
$259$	3.49942	0.217444
$260$	0	0
$261$	−0.687267	−0.0425408
$262$	0	0
$263$	−20.5661	−1.26816	−0.634081	−	0.773267i	$-0.718621\pi$
$263$	−20.5661	−1.26816	−0.634081	+	0.773267i	$0.718621\pi$
$264$	0	0
$265$	−12.9098	−0.793045
$266$	0	0
$267$	9.14958	0.559945
$268$	0	0
$269$	20.4664	1.24786	0.623930	−	0.781480i	$-0.285535\pi$
$269$	20.4664	1.24786	0.623930	+	0.781480i	$0.285535\pi$
$270$	0	0
$271$	25.1613	1.52844	0.764221	−	0.644954i	$-0.223124\pi$
$271$	25.1613	1.52844	0.764221	+	0.644954i	$0.223124\pi$
$272$	0	0
$273$	26.7546	1.61926
$274$	0	0
$275$	5.07698	0.306154
$276$	0	0
$277$	−22.8210	−1.37118	−0.685590	−	0.727987i	$-0.740456\pi$
$277$	−22.8210	−1.37118	−0.685590	+	0.727987i	$0.740456\pi$
$278$	0	0
$279$	1.66564	0.0997193
$280$	0	0
$281$	−28.9912	−1.72947	−0.864736	−	0.502227i	$-0.832514\pi$
$281$	−28.9912	−1.72947	−0.864736	+	0.502227i	$0.832514\pi$
$282$	0	0
$283$	−14.1605	−0.841756	−0.420878	−	0.907117i	$-0.638278\pi$
$283$	−14.1605	−0.841756	−0.420878	+	0.907117i	$0.638278\pi$
$284$	0	0
$285$	−5.42919	−0.321597
$286$	0	0
$287$	−16.7356	−0.987871
$288$	0	0
$289$	−15.5253	−0.913252
$290$	0	0
$291$	−5.20343	−0.305030
$292$	0	0
$293$	28.2292	1.64917	0.824583	−	0.565741i	$-0.191410\pi$
$293$	28.2292	1.64917	0.824583	+	0.565741i	$0.191410\pi$
$294$	0	0
$295$	2.47201	0.143926
$296$	0	0
$297$	27.3668	1.58798
$298$	0	0
$299$	−24.2845	−1.40441
$300$	0	0
$301$	−1.64522	−0.0948290
$302$	0	0
$303$	7.56548	0.434626
$304$	0	0
$305$	3.88479	0.222443
$306$	0	0
$307$	5.07268	0.289513	0.144756	−	0.989467i	$-0.453760\pi$
$307$	5.07268	0.289513	0.144756	+	0.989467i	$0.453760\pi$
$308$	0	0
$309$	4.55255	0.258985
$310$	0	0
$311$	13.7301	0.778564	0.389282	−	0.921119i	$-0.372723\pi$
$311$	13.7301	0.778564	0.389282	+	0.921119i	$0.372723\pi$
$312$	0	0
$313$	5.95358	0.336516	0.168258	−	0.985743i	$-0.446186\pi$
$313$	5.95358	0.336516	0.168258	+	0.985743i	$0.446186\pi$
$314$	0	0
$315$	−1.03143	−0.0581145
$316$	0	0
$317$	29.6043	1.66274	0.831372	−	0.555716i	$-0.187556\pi$
$317$	29.6043	1.66274	0.831372	+	0.555716i	$0.187556\pi$
$318$	0	0
$319$	13.8870	0.777521
$320$	0	0
$321$	4.34368	0.242441
$322$	0	0
$323$	−3.97669	−0.221269
$324$	0	0
$325$	−3.93111	−0.218059
$326$	0	0
$327$	12.6616	0.700190
$328$	0	0
$329$	34.5927	1.90716
$330$	0	0
$331$	−19.9511	−1.09661	−0.548305	−	0.836279i	$-0.684727\pi$
$331$	−19.9511	−1.09661	−0.548305	+	0.836279i	$0.684727\pi$
$332$	0	0
$333$	−0.214192	−0.0117377
$334$	0	0
$335$	10.2444	0.559713
$336$	0	0
$337$	−3.64207	−0.198396	−0.0991981	−	0.995068i	$-0.531628\pi$
$337$	−3.64207	−0.198396	−0.0991981	+	0.995068i	$0.531628\pi$
$338$	0	0
$339$	0.747304	0.0405880
$340$	0	0
$341$	−33.6561	−1.82258
$342$	0	0
$343$	11.7044	0.631980
$344$	0	0
$345$	−10.2419	−0.551407
$346$	0	0
$347$	9.47547	0.508670	0.254335	−	0.967116i	$-0.418143\pi$
$347$	9.47547	0.508670	0.254335	+	0.967116i	$0.418143\pi$
$348$	0	0
$349$	8.46218	0.452970	0.226485	−	0.974015i	$-0.427277\pi$
$349$	8.46218	0.452970	0.226485	+	0.974015i	$0.427277\pi$
$350$	0	0
$351$	−21.1901	−1.13105
$352$	0	0
$353$	8.62160	0.458882	0.229441	−	0.973323i	$-0.426310\pi$
$353$	8.62160	0.458882	0.229441	+	0.973323i	$0.426310\pi$
$354$	0	0
$355$	8.58441	0.455613
$356$	0	0
$357$	8.26489	0.437424
$358$	0	0
$359$	17.1941	0.907469	0.453734	−	0.891137i	$-0.350091\pi$
$359$	17.1941	0.907469	0.453734	+	0.891137i	$0.350091\pi$
$360$	0	0
$361$	−8.27651	−0.435606
$362$	0	0
$363$	−24.4972	−1.28577
$364$	0	0
$365$	5.83210	0.305266
$366$	0	0
$367$	0.886421	0.0462708	0.0231354	−	0.999732i	$-0.492635\pi$
$367$	0.886421	0.0462708	0.0231354	+	0.999732i	$0.492635\pi$
$368$	0	0
$369$	1.02435	0.0533256
$370$	0	0
$371$	−52.9952	−2.75137
$372$	0	0
$373$	0.960747	0.0497456	0.0248728	−	0.999691i	$-0.492082\pi$
$373$	0.960747	0.0497456	0.0248728	+	0.999691i	$0.492082\pi$
$374$	0	0
$375$	−1.65793	−0.0856153
$376$	0	0
$377$	−10.7527	−0.553792
$378$	0	0
$379$	13.0321	0.669412	0.334706	−	0.942323i	$-0.391363\pi$
$379$	13.0321	0.669412	0.334706	+	0.942323i	$0.391363\pi$
$380$	0	0
$381$	−8.41192	−0.430956
$382$	0	0
$383$	−5.73956	−0.293278	−0.146639	−	0.989190i	$-0.546846\pi$
$383$	−5.73956	−0.293278	−0.146639	+	0.989190i	$0.546846\pi$
$384$	0	0
$385$	20.8412	1.06216
$386$	0	0
$387$	0.100701	0.00511890
$388$	0	0
$389$	7.07243	0.358587	0.179293	−	0.983796i	$-0.442619\pi$
$389$	7.07243	0.358587	0.179293	+	0.983796i	$0.442619\pi$
$390$	0	0
$391$	−7.50186	−0.379385
$392$	0	0
$393$	16.7628	0.845573
$394$	0	0
$395$	9.07348	0.456537
$396$	0	0
$397$	0.290288	0.0145691	0.00728456	−	0.999973i	$-0.497681\pi$
$397$	0.290288	0.0145691	0.00728456	+	0.999973i	$0.497681\pi$
$398$	0	0
$399$	−22.2870	−1.11574
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	26.0599	1.29814
$404$	0	0
$405$	−8.18309	−0.406621
$406$	0	0
$407$	4.32799	0.214531
$408$	0	0
$409$	−5.37493	−0.265773	−0.132887	−	0.991131i	$-0.542425\pi$
$409$	−5.37493	−0.265773	−0.132887	+	0.991131i	$0.542425\pi$
$410$	0	0
$411$	−0.333631	−0.0164568
$412$	0	0
$413$	10.1477	0.499334
$414$	0	0
$415$	5.00895	0.245880
$416$	0	0
$417$	−19.5805	−0.958863
$418$	0	0
$419$	−31.7315	−1.55019	−0.775093	−	0.631848i	$-0.782297\pi$
$419$	−31.7315	−1.55019	−0.775093	+	0.631848i	$0.782297\pi$
$420$	0	0
$421$	22.5550	1.09926	0.549632	−	0.835407i	$-0.314768\pi$
$421$	22.5550	1.09926	0.549632	+	0.835407i	$0.314768\pi$
$422$	0	0
$423$	−2.11735	−0.102949
$424$	0	0
$425$	−1.21438	−0.0589060
$426$	0	0
$427$	15.9472	0.771738
$428$	0	0
$429$	33.0893	1.59757
$430$	0	0
$431$	0.0532839	0.00256660	0.00128330	−	0.999999i	$-0.499592\pi$
$431$	0.0532839	0.00256660	0.00128330	+	0.999999i	$0.499592\pi$
$432$	0	0
$433$	33.7006	1.61955	0.809773	−	0.586744i	$-0.199590\pi$
$433$	33.7006	1.61955	0.809773	+	0.586744i	$0.199590\pi$
$434$	0	0
$435$	−4.53491	−0.217432
$436$	0	0
$437$	20.2294	0.967703
$438$	0	0
$439$	−13.3265	−0.636039	−0.318019	−	0.948084i	$-0.603018\pi$
$439$	−13.3265	−0.636039	−0.318019	+	0.948084i	$0.603018\pi$
$440$	0	0
$441$	−2.47522	−0.117868
$442$	0	0
$443$	31.7170	1.50692	0.753460	−	0.657494i	$-0.228384\pi$
$443$	31.7170	1.50692	0.753460	+	0.657494i	$0.228384\pi$
$444$	0	0
$445$	−5.51867	−0.261610
$446$	0	0
$447$	12.7052	0.600934
$448$	0	0
$449$	23.6272	1.11503	0.557517	−	0.830165i	$-0.311754\pi$
$449$	23.6272	1.11503	0.557517	+	0.830165i	$0.311754\pi$
$450$	0	0
$451$	−20.6981	−0.974637
$452$	0	0
$453$	6.86336	0.322469
$454$	0	0
$455$	−16.1373	−0.756529
$456$	0	0
$457$	12.8086	0.599159	0.299580	−	0.954071i	$-0.403154\pi$
$457$	12.8086	0.599159	0.299580	+	0.954071i	$0.403154\pi$
$458$	0	0
$459$	−6.54595	−0.305539
$460$	0	0
$461$	−6.27555	−0.292282	−0.146141	−	0.989264i	$-0.546685\pi$
$461$	−6.27555	−0.292282	−0.146141	+	0.989264i	$0.546685\pi$
$462$	0	0
$463$	32.1767	1.49538	0.747690	−	0.664048i	$-0.231163\pi$
$463$	32.1767	1.49538	0.747690	+	0.664048i	$0.231163\pi$
$464$	0	0
$465$	10.9907	0.509681
$466$	0	0
$467$	0.324866	0.0150330	0.00751650	−	0.999972i	$-0.497607\pi$
$467$	0.324866	0.0150330	0.00751650	+	0.999972i	$0.497607\pi$
$468$	0	0
$469$	42.0537	1.94186
$470$	0	0
$471$	−2.76839	−0.127561
$472$	0	0
$473$	−2.03476	−0.0935585
$474$	0	0
$475$	3.27467	0.150252
$476$	0	0
$477$	3.24373	0.148520
$478$	0	0
$479$	35.0330	1.60070	0.800349	−	0.599534i	$-0.204648\pi$
$479$	35.0330	1.60070	0.800349	+	0.599534i	$0.204648\pi$
$480$	0	0
$481$	−3.35117	−0.152800
$482$	0	0
$483$	−42.0434	−1.91304
$484$	0	0
$485$	3.13851	0.142512
$486$	0	0
$487$	12.6993	0.575462	0.287731	−	0.957711i	$-0.407099\pi$
$487$	12.6993	0.575462	0.287731	+	0.957711i	$0.407099\pi$
$488$	0	0
$489$	−20.5024	−0.927149
$490$	0	0
$491$	32.2739	1.45650	0.728251	−	0.685310i	$-0.240333\pi$
$491$	32.2739	1.45650	0.728251	+	0.685310i	$0.240333\pi$
$492$	0	0
$493$	−3.32167	−0.149600
$494$	0	0
$495$	−1.27564	−0.0573359
$496$	0	0
$497$	35.2392	1.58070
$498$	0	0
$499$	−3.81648	−0.170849	−0.0854246	−	0.996345i	$-0.527225\pi$
$499$	−3.81648	−0.170849	−0.0854246	+	0.996345i	$0.527225\pi$
$500$	0	0
$501$	19.2315	0.859200
$502$	0	0
$503$	23.8084	1.06156	0.530782	−	0.847508i	$-0.321898\pi$
$503$	23.8084	1.06156	0.530782	+	0.847508i	$0.321898\pi$
$504$	0	0
$505$	−4.56320	−0.203060
$506$	0	0
$507$	−4.06795	−0.180664
$508$	0	0
$509$	−16.1170	−0.714373	−0.357186	−	0.934033i	$-0.616264\pi$
$509$	−16.1170	−0.714373	−0.357186	+	0.934033i	$0.616264\pi$
$510$	0	0
$511$	23.9409	1.05908
$512$	0	0
$513$	17.6517	0.779342
$514$	0	0
$515$	−2.74592	−0.121000
$516$	0	0
$517$	42.7833	1.88161
$518$	0	0
$519$	0.0896261	0.00393415
$520$	0	0
$521$	21.7731	0.953896	0.476948	−	0.878931i	$-0.341743\pi$
$521$	21.7731	0.953896	0.476948	+	0.878931i	$0.341743\pi$
$522$	0	0
$523$	−19.9746	−0.873430	−0.436715	−	0.899600i	$-0.643858\pi$
$523$	−19.9746	−0.873430	−0.436715	+	0.899600i	$0.643858\pi$
$524$	0	0
$525$	−6.80586	−0.297032
$526$	0	0
$527$	8.05030	0.350677
$528$	0	0
$529$	15.1619	0.659211
$530$	0	0
$531$	−0.621118	−0.0269542
$532$	0	0
$533$	16.0266	0.694188
$534$	0	0
$535$	−2.61994	−0.113270
$536$	0	0
$537$	37.0262	1.59780
$538$	0	0
$539$	50.0146	2.15428
$540$	0	0
$541$	−0.0536480	−0.00230651	−0.00115325	−	0.999999i	$-0.500367\pi$
$541$	−0.0536480	−0.00230651	−0.00115325	+	0.999999i	$0.500367\pi$
$542$	0	0
$543$	29.6526	1.27251
$544$	0	0
$545$	−7.63700	−0.327133
$546$	0	0
$547$	−1.86662	−0.0798108	−0.0399054	−	0.999203i	$-0.512706\pi$
$547$	−1.86662	−0.0798108	−0.0399054	+	0.999203i	$0.512706\pi$
$548$	0	0
$549$	−0.976094	−0.0416587
$550$	0	0
$551$	8.95715	0.381588
$552$	0	0
$553$	37.2469	1.58390
$554$	0	0
$555$	−1.41334	−0.0599931
$556$	0	0
$557$	23.3365	0.988801	0.494401	−	0.869234i	$-0.335388\pi$
$557$	23.3365	0.988801	0.494401	+	0.869234i	$0.335388\pi$
$558$	0	0
$559$	1.57552	0.0666373
$560$	0	0
$561$	10.2218	0.431564
$562$	0	0
$563$	−23.2839	−0.981298	−0.490649	−	0.871357i	$-0.663240\pi$
$563$	−23.2839	−0.981298	−0.490649	+	0.871357i	$0.663240\pi$
$564$	0	0
$565$	−0.450744	−0.0189630
$566$	0	0
$567$	−33.5918	−1.41072
$568$	0	0
$569$	34.8727	1.46194	0.730970	−	0.682409i	$-0.239068\pi$
$569$	34.8727	1.46194	0.730970	+	0.682409i	$0.239068\pi$
$570$	0	0
$571$	0.344308	0.0144089	0.00720443	−	0.999974i	$-0.497707\pi$
$571$	0.344308	0.0144089	0.00720443	+	0.999974i	$0.497707\pi$
$572$	0	0
$573$	−1.88045	−0.0785571
$574$	0	0
$575$	6.17753	0.257621
$576$	0	0
$577$	43.8682	1.82626	0.913129	−	0.407672i	$-0.133659\pi$
$577$	43.8682	1.82626	0.913129	+	0.407672i	$0.133659\pi$
$578$	0	0
$579$	−19.3668	−0.804858
$580$	0	0
$581$	20.5619	0.853051
$582$	0	0
$583$	−65.5430	−2.71451
$584$	0	0
$585$	0.987731	0.0408377
$586$	0	0
$587$	5.36434	0.221410	0.110705	−	0.993853i	$-0.464689\pi$
$587$	5.36434	0.221410	0.110705	+	0.993853i	$0.464689\pi$
$588$	0	0
$589$	−21.7083	−0.894475
$590$	0	0
$591$	−26.7849	−1.10178
$592$	0	0
$593$	5.11417	0.210014	0.105007	−	0.994472i	$-0.466514\pi$
$593$	5.11417	0.210014	0.105007	+	0.994472i	$0.466514\pi$
$594$	0	0
$595$	−4.98506	−0.204367
$596$	0	0
$597$	−13.3755	−0.547421
$598$	0	0
$599$	43.0937	1.76076	0.880380	−	0.474269i	$-0.157287\pi$
$599$	43.0937	1.76076	0.880380	+	0.474269i	$0.157287\pi$
$600$	0	0
$601$	4.33725	0.176920	0.0884601	−	0.996080i	$-0.471805\pi$
$601$	4.33725	0.176920	0.0884601	+	0.996080i	$0.471805\pi$
$602$	0	0
$603$	−2.57402	−0.104822
$604$	0	0
$605$	14.7758	0.600720
$606$	0	0
$607$	−5.93703	−0.240976	−0.120488	−	0.992715i	$-0.538446\pi$
$607$	−5.93703	−0.240976	−0.120488	+	0.992715i	$0.538446\pi$
$608$	0	0
$609$	−18.6159	−0.754356
$610$	0	0
$611$	−33.1271	−1.34018
$612$	0	0
$613$	13.9837	0.564796	0.282398	−	0.959297i	$-0.408870\pi$
$613$	13.9837	0.564796	0.282398	+	0.959297i	$0.408870\pi$
$614$	0	0
$615$	6.75915	0.272555
$616$	0	0
$617$	8.88037	0.357510	0.178755	−	0.983894i	$-0.442793\pi$
$617$	8.88037	0.357510	0.178755	+	0.983894i	$0.442793\pi$
$618$	0	0
$619$	−22.3196	−0.897101	−0.448550	−	0.893758i	$-0.648060\pi$
$619$	−22.3196	−0.897101	−0.448550	+	0.893758i	$0.648060\pi$
$620$	0	0
$621$	33.2992	1.33625
$622$	0	0
$623$	−22.6543	−0.907625
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−27.5639	−1.10080
$628$	0	0
$629$	−1.03523	−0.0412771
$630$	0	0
$631$	−10.0509	−0.400121	−0.200060	−	0.979784i	$-0.564114\pi$
$631$	−10.0509	−0.400121	−0.200060	+	0.979784i	$0.564114\pi$
$632$	0	0
$633$	−37.9993	−1.51034
$634$	0	0
$635$	5.07374	0.201345
$636$	0	0
$637$	−38.7263	−1.53439
$638$	0	0
$639$	−2.15692	−0.0853265
$640$	0	0
$641$	−45.6382	−1.80260	−0.901300	−	0.433195i	$-0.857386\pi$
$641$	−45.6382	−1.80260	−0.901300	+	0.433195i	$0.857386\pi$
$642$	0	0
$643$	−7.05551	−0.278242	−0.139121	−	0.990275i	$-0.544428\pi$
$643$	−7.05551	−0.278242	−0.139121	+	0.990275i	$0.544428\pi$
$644$	0	0
$645$	0.664470	0.0261635
$646$	0	0
$647$	20.7372	0.815262	0.407631	−	0.913147i	$-0.366355\pi$
$647$	20.7372	0.815262	0.407631	+	0.913147i	$0.366355\pi$
$648$	0	0
$649$	12.5504	0.492645
$650$	0	0
$651$	45.1171	1.76828
$652$	0	0
$653$	−16.5336	−0.647009	−0.323505	−	0.946227i	$-0.604861\pi$
$653$	−16.5336	−0.647009	−0.323505	+	0.946227i	$0.604861\pi$
$654$	0	0
$655$	−10.1107	−0.395057
$656$	0	0
$657$	−1.46537	−0.0571697
$658$	0	0
$659$	−17.3940	−0.677576	−0.338788	−	0.940863i	$-0.610017\pi$
$659$	−17.3940	−0.677576	−0.338788	+	0.940863i	$0.610017\pi$
$660$	0	0
$661$	−21.2874	−0.827983	−0.413992	−	0.910281i	$-0.635866\pi$
$661$	−21.2874	−0.827983	−0.413992	+	0.910281i	$0.635866\pi$
$662$	0	0
$663$	−7.91473	−0.307383
$664$	0	0
$665$	13.4426	0.521283
$666$	0	0
$667$	16.8973	0.654265
$668$	0	0
$669$	12.9096	0.499115
$670$	0	0
$671$	19.7230	0.761399
$672$	0	0
$673$	14.6902	0.566266	0.283133	−	0.959081i	$-0.408626\pi$
$673$	14.6902	0.566266	0.283133	+	0.959081i	$0.408626\pi$
$674$	0	0
$675$	5.39037	0.207475
$676$	0	0
$677$	−22.9321	−0.881353	−0.440676	−	0.897666i	$-0.645261\pi$
$677$	−22.9321	−0.881353	−0.440676	+	0.897666i	$0.645261\pi$
$678$	0	0
$679$	12.8836	0.494429
$680$	0	0
$681$	22.9411	0.879104
$682$	0	0
$683$	32.8221	1.25590	0.627951	−	0.778253i	$-0.283894\pi$
$683$	32.8221	1.25590	0.627951	+	0.778253i	$0.283894\pi$
$684$	0	0
$685$	0.201233	0.00768873
$686$	0	0
$687$	−4.24733	−0.162046
$688$	0	0
$689$	50.7500	1.93342
$690$	0	0
$691$	47.8669	1.82094	0.910472	−	0.413570i	$-0.135718\pi$
$691$	47.8669	1.82094	0.910472	+	0.413570i	$0.135718\pi$
$692$	0	0
$693$	−5.23655	−0.198920
$694$	0	0
$695$	11.8102	0.447987
$696$	0	0
$697$	4.95085	0.187527
$698$	0	0
$699$	−3.91821	−0.148200
$700$	0	0
$701$	25.9036	0.978365	0.489183	−	0.872181i	$-0.337295\pi$
$701$	25.9036	0.978365	0.489183	+	0.872181i	$0.337295\pi$
$702$	0	0
$703$	2.79157	0.105286
$704$	0	0
$705$	−13.9713	−0.526188
$706$	0	0
$707$	−18.7321	−0.704492
$708$	0	0
$709$	−19.1750	−0.720133	−0.360066	−	0.932927i	$-0.617246\pi$
$709$	−19.1750	−0.720133	−0.360066	+	0.932927i	$0.617246\pi$
$710$	0	0
$711$	−2.27980	−0.0854994
$712$	0	0
$713$	−40.9518	−1.53366
$714$	0	0
$715$	−19.9582	−0.746394
$716$	0	0
$717$	−19.2639	−0.719424
$718$	0	0
$719$	−3.53916	−0.131988	−0.0659941	−	0.997820i	$-0.521022\pi$
$719$	−3.53916	−0.131988	−0.0659941	+	0.997820i	$0.521022\pi$
$720$	0	0
$721$	−11.2721	−0.419794
$722$	0	0
$723$	45.9398	1.70852
$724$	0	0
$725$	2.73528	0.101586
$726$	0	0
$727$	−36.7027	−1.36123	−0.680614	−	0.732642i	$-0.738287\pi$
$727$	−36.7027	−1.36123	−0.680614	+	0.732642i	$0.738287\pi$
$728$	0	0
$729$	28.8667	1.06914
$730$	0	0
$731$	0.486701	0.0180013
$732$	0	0
$733$	−10.8575	−0.401031	−0.200515	−	0.979691i	$-0.564262\pi$
$733$	−10.8575	−0.401031	−0.200515	+	0.979691i	$0.564262\pi$
$734$	0	0
$735$	−16.3327	−0.602441
$736$	0	0
$737$	52.0108	1.91584
$738$	0	0
$739$	−26.7210	−0.982949	−0.491475	−	0.870892i	$-0.663542\pi$
$739$	−26.7210	−0.982949	−0.491475	+	0.870892i	$0.663542\pi$
$740$	0	0
$741$	21.3427	0.784045
$742$	0	0
$743$	20.5017	0.752134	0.376067	−	0.926592i	$-0.377276\pi$
$743$	20.5017	0.752134	0.376067	+	0.926592i	$0.377276\pi$
$744$	0	0
$745$	−7.66326	−0.280760
$746$	0	0
$747$	−1.25855	−0.0460479
$748$	0	0
$749$	−10.7549	−0.392976
$750$	0	0
$751$	−53.3760	−1.94772	−0.973859	−	0.227152i	$-0.927059\pi$
$751$	−53.3760	−1.94772	−0.973859	+	0.227152i	$0.927059\pi$
$752$	0	0
$753$	7.03820	0.256486
$754$	0	0
$755$	−4.13971	−0.150659
$756$	0	0
$757$	−33.2057	−1.20688	−0.603441	−	0.797408i	$-0.706204\pi$
$757$	−33.2057	−1.20688	−0.603441	+	0.797408i	$0.706204\pi$
$758$	0	0
$759$	−51.9981	−1.88741
$760$	0	0
$761$	11.8221	0.428550	0.214275	−	0.976773i	$-0.431261\pi$
$761$	11.8221	0.428550	0.214275	+	0.976773i	$0.431261\pi$
$762$	0	0
$763$	−31.3501	−1.13495
$764$	0	0
$765$	0.305125	0.0110318
$766$	0	0
$767$	−9.71775	−0.350887
$768$	0	0
$769$	−38.3269	−1.38210	−0.691051	−	0.722806i	$-0.742852\pi$
$769$	−38.3269	−1.38210	−0.691051	+	0.722806i	$0.742852\pi$
$770$	0	0
$771$	−6.90874	−0.248812
$772$	0	0
$773$	−26.1490	−0.940515	−0.470257	−	0.882529i	$-0.655839\pi$
$773$	−26.1490	−0.940515	−0.470257	+	0.882529i	$0.655839\pi$
$774$	0	0
$775$	−6.62915	−0.238126
$776$	0	0
$777$	−5.80181	−0.208139
$778$	0	0
$779$	−13.3504	−0.478327
$780$	0	0
$781$	43.5829	1.55952
$782$	0	0
$783$	14.7442	0.526914
$784$	0	0
$785$	1.66979	0.0595972
$786$	0	0
$787$	−32.6279	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$787$	−32.6279	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$788$	0	0
$789$	34.0972	1.21389
$790$	0	0
$791$	−1.85032	−0.0657897
$792$	0	0
$793$	−15.2716	−0.542309
$794$	0	0
$795$	21.4036	0.759109
$796$	0	0
$797$	37.2820	1.32060	0.660299	−	0.751003i	$-0.270430\pi$
$797$	37.2820	1.32060	0.660299	+	0.751003i	$0.270430\pi$
$798$	0	0
$799$	−10.2335	−0.362034
$800$	0	0
$801$	1.38662	0.0489939
$802$	0	0
$803$	29.6095	1.04490
$804$	0	0
$805$	25.3589	0.893784
$806$	0	0
$807$	−33.9320	−1.19446
$808$	0	0
$809$	−0.706997	−0.0248567	−0.0124283	−	0.999923i	$-0.503956\pi$
$809$	−0.706997	−0.0248567	−0.0124283	+	0.999923i	$0.503956\pi$
$810$	0	0
$811$	1.25708	0.0441420	0.0220710	−	0.999756i	$-0.492974\pi$
$811$	1.25708	0.0441420	0.0220710	+	0.999756i	$0.492974\pi$
$812$	0	0
$813$	−41.7158	−1.46304
$814$	0	0
$815$	12.3662	0.433170
$816$	0	0
$817$	−1.31243	−0.0459161
$818$	0	0
$819$	4.05466	0.141681
$820$	0	0
$821$	−15.6783	−0.547175	−0.273588	−	0.961847i	$-0.588210\pi$
$821$	−15.6783	−0.547175	−0.273588	+	0.961847i	$0.588210\pi$
$822$	0	0
$823$	−36.6447	−1.27736	−0.638678	−	0.769474i	$-0.720518\pi$
$823$	−36.6447	−1.27736	−0.638678	+	0.769474i	$0.720518\pi$
$824$	0	0
$825$	−8.41730	−0.293053
$826$	0	0
$827$	40.1587	1.39645	0.698227	−	0.715877i	$-0.253973\pi$
$827$	40.1587	1.39645	0.698227	+	0.715877i	$0.253973\pi$
$828$	0	0
$829$	8.70334	0.302279	0.151140	−	0.988512i	$-0.451706\pi$
$829$	8.70334	0.302279	0.151140	+	0.988512i	$0.451706\pi$
$830$	0	0
$831$	37.8357	1.31250
$832$	0	0
$833$	−11.9631	−0.414498
$834$	0	0
$835$	−11.5997	−0.401424
$836$	0	0
$837$	−35.7336	−1.23513
$838$	0	0
$839$	−11.6051	−0.400653	−0.200326	−	0.979729i	$-0.564200\pi$
$839$	−11.6051	−0.400653	−0.200326	+	0.979729i	$0.564200\pi$
$840$	0	0
$841$	−21.5182	−0.742008
$842$	0	0
$843$	48.0655	1.65546
$844$	0	0
$845$	2.45363	0.0844073
$846$	0	0
$847$	60.6549	2.08413
$848$	0	0
$849$	23.4772	0.805735
$850$	0	0
$851$	5.26618	0.180522
$852$	0	0
$853$	−41.1577	−1.40921	−0.704607	−	0.709598i	$-0.748877\pi$
$853$	−41.1577	−1.40921	−0.704607	+	0.709598i	$0.748877\pi$
$854$	0	0
$855$	−0.822795	−0.0281390
$856$	0	0
$857$	30.1478	1.02983	0.514915	−	0.857241i	$-0.327823\pi$
$857$	30.1478	1.02983	0.514915	+	0.857241i	$0.327823\pi$
$858$	0	0
$859$	3.53796	0.120714	0.0603569	−	0.998177i	$-0.480776\pi$
$859$	3.53796	0.120714	0.0603569	+	0.998177i	$0.480776\pi$
$860$	0	0
$861$	27.7465	0.945598
$862$	0	0
$863$	34.7222	1.18196	0.590978	−	0.806687i	$-0.298742\pi$
$863$	34.7222	1.18196	0.590978	+	0.806687i	$0.298742\pi$
$864$	0	0
$865$	−0.0540590	−0.00183806
$866$	0	0
$867$	25.7399	0.874172
$868$	0	0
$869$	46.0659	1.56268
$870$	0	0
$871$	−40.2720	−1.36456
$872$	0	0
$873$	−0.788581	−0.0266894
$874$	0	0
$875$	4.10503	0.138775
$876$	0	0
$877$	−15.0891	−0.509522	−0.254761	−	0.967004i	$-0.581997\pi$
$877$	−15.0891	−0.509522	−0.254761	+	0.967004i	$0.581997\pi$
$878$	0	0
$879$	−46.8021	−1.57860
$880$	0	0
$881$	−3.55023	−0.119610	−0.0598051	−	0.998210i	$-0.519048\pi$
$881$	−3.55023	−0.119610	−0.0598051	+	0.998210i	$0.519048\pi$
$882$	0	0
$883$	53.5178	1.80102	0.900509	−	0.434837i	$-0.143194\pi$
$883$	53.5178	1.80102	0.900509	+	0.434837i	$0.143194\pi$
$884$	0	0
$885$	−4.09843	−0.137767
$886$	0	0
$887$	45.8102	1.53816	0.769078	−	0.639155i	$-0.220716\pi$
$887$	45.8102	1.53816	0.769078	+	0.639155i	$0.220716\pi$
$888$	0	0
$889$	20.8278	0.698543
$890$	0	0
$891$	−41.5454	−1.39182
$892$	0	0
$893$	27.5954	0.923444
$894$	0	0
$895$	−22.3328	−0.746502
$896$	0	0
$897$	40.2621	1.34431
$898$	0	0
$899$	−18.1326	−0.604756
$900$	0	0
$901$	15.6774	0.522291
$902$	0	0
$903$	2.72767	0.0907710
$904$	0	0
$905$	−17.8853	−0.594527
$906$	0	0
$907$	3.38627	0.112439	0.0562197	−	0.998418i	$-0.482095\pi$
$907$	3.38627	0.112439	0.0562197	+	0.998418i	$0.482095\pi$
$908$	0	0
$909$	1.14655	0.0380287
$910$	0	0
$911$	39.9233	1.32272	0.661359	−	0.750070i	$-0.269980\pi$
$911$	39.9233	1.32272	0.661359	+	0.750070i	$0.269980\pi$
$912$	0	0
$913$	25.4304	0.841623
$914$	0	0
$915$	−6.44073	−0.212924
$916$	0	0
$917$	−41.5046	−1.37060
$918$	0	0
$919$	−4.27484	−0.141014	−0.0705070	−	0.997511i	$-0.522462\pi$
$919$	−4.27484	−0.141014	−0.0705070	+	0.997511i	$0.522462\pi$
$920$	0	0
$921$	−8.41015	−0.277124
$922$	0	0
$923$	−33.7463	−1.11077
$924$	0	0
$925$	0.852473	0.0280291
$926$	0	0
$927$	0.689940	0.0226606
$928$	0	0
$929$	−4.42260	−0.145101	−0.0725504	−	0.997365i	$-0.523114\pi$
$929$	−4.42260	−0.145101	−0.0725504	+	0.997365i	$0.523114\pi$
$930$	0	0
$931$	32.2596	1.05727
$932$	0	0
$933$	−22.7636	−0.745248
$934$	0	0
$935$	−6.16538	−0.201630
$936$	0	0
$937$	−36.5581	−1.19430	−0.597151	−	0.802129i	$-0.703701\pi$
$937$	−36.5581	−1.19430	−0.597151	+	0.802129i	$0.703701\pi$
$938$	0	0
$939$	−9.87063	−0.322116
$940$	0	0
$941$	21.8709	0.712970	0.356485	−	0.934301i	$-0.383975\pi$
$941$	21.8709	0.712970	0.356485	+	0.934301i	$0.383975\pi$
$942$	0	0
$943$	−25.1849	−0.820133
$944$	0	0
$945$	22.1276	0.719811
$946$	0	0
$947$	−20.1218	−0.653869	−0.326935	−	0.945047i	$-0.606016\pi$
$947$	−20.1218	−0.653869	−0.326935	+	0.945047i	$0.606016\pi$
$948$	0	0
$949$	−22.9266	−0.744230
$950$	0	0
$951$	−49.0820	−1.59159
$952$	0	0
$953$	6.49432	0.210372	0.105186	−	0.994453i	$-0.466456\pi$
$953$	6.49432	0.210372	0.105186	+	0.994453i	$0.466456\pi$
$954$	0	0
$955$	1.13422	0.0367024
$956$	0	0
$957$	−23.0237	−0.744250
$958$	0	0
$959$	0.826068	0.0266751
$960$	0	0
$961$	12.9456	0.417602
$962$	0	0
$963$	0.658286	0.0212130
$964$	0	0
$965$	11.6813	0.376035
$966$	0	0
$967$	−21.8867	−0.703830	−0.351915	−	0.936032i	$-0.614469\pi$
$967$	−21.8867	−0.703830	−0.351915	+	0.936032i	$0.614469\pi$
$968$	0	0
$969$	6.59309	0.211801
$970$	0	0
$971$	−28.8521	−0.925907	−0.462953	−	0.886383i	$-0.653210\pi$
$971$	−28.8521	−0.925907	−0.462953	+	0.886383i	$0.653210\pi$
$972$	0	0
$973$	48.4812	1.55424
$974$	0	0
$975$	6.51751	0.208728
$976$	0	0
$977$	−20.4172	−0.653206	−0.326603	−	0.945162i	$-0.605904\pi$
$977$	−20.4172	−0.653206	−0.326603	+	0.945162i	$0.605904\pi$
$978$	0	0
$979$	−28.0182	−0.895465
$980$	0	0
$981$	1.91887	0.0612649
$982$	0	0
$983$	−16.8100	−0.536157	−0.268079	−	0.963397i	$-0.586389\pi$
$983$	−16.8100	−0.536157	−0.268079	+	0.963397i	$0.586389\pi$
$984$	0	0
$985$	16.1556	0.514760
$986$	0	0
$987$	−57.3524	−1.82555
$988$	0	0
$989$	−2.47584	−0.0787272
$990$	0	0
$991$	−5.89608	−0.187295	−0.0936475	−	0.995605i	$-0.529853\pi$
$991$	−5.89608	−0.187295	−0.0936475	+	0.995605i	$0.529853\pi$
$992$	0	0
$993$	33.0775	1.04968
$994$	0	0
$995$	8.06755	0.255759
$996$	0	0
$997$	30.3436	0.960991	0.480496	−	0.876997i	$-0.340457\pi$
$997$	30.3436	0.960991	0.480496	+	0.876997i	$0.340457\pi$
$998$	0	0
$999$	4.59514	0.145384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.11	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.11	✓	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65793 q^{3} +1.00000 q^{5} +4.10503 q^{7} -0.251260 q^{9} +O(q^{10})\)	\(q-1.65793 q^{3} +1.00000 q^{5} +4.10503 q^{7} -0.251260 q^{9} +5.07698 q^{11} -3.93111 q^{13} -1.65793 q^{15} -1.21438 q^{17} +3.27467 q^{19} -6.80586 q^{21} +6.17753 q^{23} +1.00000 q^{25} +5.39037 q^{27} +2.73528 q^{29} -6.62915 q^{31} -8.41730 q^{33} +4.10503 q^{35} +0.852473 q^{37} +6.51751 q^{39} -4.07686 q^{41} -0.400782 q^{43} -0.251260 q^{45} +8.42691 q^{47} +9.85125 q^{49} +2.01336 q^{51} -12.9098 q^{53} +5.07698 q^{55} -5.42919 q^{57} +2.47201 q^{59} +3.88479 q^{61} -1.03143 q^{63} -3.93111 q^{65} +10.2444 q^{67} -10.2419 q^{69} +8.58441 q^{71} +5.83210 q^{73} -1.65793 q^{75} +20.8412 q^{77} +9.07348 q^{79} -8.18309 q^{81} +5.00895 q^{83} -1.21438 q^{85} -4.53491 q^{87} -5.51867 q^{89} -16.1373 q^{91} +10.9907 q^{93} +3.27467 q^{95} +3.13851 q^{97} -1.27564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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