LMFDB - Embedded newform 10000.2.a.bq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80 $ x^16 - 4x^15 - 26x^14 + 110x^13 + 250x^12 - 1154x^11 - 1074x^10 + 5784x^9 + 1890x^8 - 14210x^7 - 726x^6 + 15974x^5 + 61x^4 - 7820x^3 - 180x^2 + 1360*x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.30769$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30769 q^{3} -4.74404 q^{7} +2.32542 q^{9} +5.47714 q^{11} +0.899642 q^{13} -0.714260 q^{17} -5.35412 q^{19} +10.9478 q^{21} +3.04153 q^{23} +1.55671 q^{27} -0.636866 q^{29} -6.03862 q^{31} -12.6395 q^{33} -1.69746 q^{37} -2.07609 q^{39} +10.3516 q^{41} -8.11200 q^{43} -0.909166 q^{47} +15.5059 q^{49} +1.64829 q^{51} -1.02940 q^{53} +12.3556 q^{57} -5.59978 q^{59} +8.18481 q^{61} -11.0319 q^{63} -1.36315 q^{67} -7.01890 q^{69} -1.56534 q^{71} +8.11080 q^{73} -25.9837 q^{77} +12.1629 q^{79} -10.5687 q^{81} -12.8315 q^{83} +1.46969 q^{87} -3.98523 q^{89} -4.26793 q^{91} +13.9353 q^{93} -8.95131 q^{97} +12.7367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + 12 q^{11} + 10 q^{13} + 8 q^{17} + 12 q^{19} + 8 q^{21} - 12 q^{23} - 22 q^{27} + 16 q^{29} + 2 q^{31} + 24 q^{33} + 22 q^{37} + 4 q^{39} + 20 q^{41} - 26 q^{43} - 24 q^{47}+ \cdots + 46 q^{99}+O(q^{100}) $ 16 * q - 4 * q^3 - 8 * q^7 + 20 * q^9 + 12 * q^11 + 10 * q^13 + 8 * q^17 + 12 * q^19 + 8 * q^21 - 12 * q^23 - 22 * q^27 + 16 * q^29 + 2 * q^31 + 24 * q^33 + 22 * q^37 + 4 * q^39 + 20 * q^41 - 26 * q^43 - 24 * q^47 + 30 * q^49 + 30 * q^51 + 16 * q^53 + 24 * q^57 + 40 * q^59 + 22 * q^61 - 52 * q^63 - 50 * q^67 + 22 * q^69 + 4 * q^71 + 32 * q^73 + 30 * q^77 + 8 * q^79 + 44 * q^81 - 30 * q^83 - 48 * q^87 + 28 * q^89 + 32 * q^91 + 40 * q^93 + 36 * q^97 + 46 * 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
$3$	−2.30769	−1.33234	−0.666172	−	0.745798i	$-0.732068\pi$
$3$	−2.30769	−1.33234	−0.666172	+	0.745798i	$0.732068\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.74404	−1.79308	−0.896538	−	0.442966i	$-0.853926\pi$
$7$	−4.74404	−1.79308	−0.896538	+	0.442966i	$0.853926\pi$
$8$	0	0
$9$	2.32542	0.775141
$10$	0	0
$11$	5.47714	1.65142	0.825710	−	0.564095i	$-0.190775\pi$
$11$	5.47714	1.65142	0.825710	+	0.564095i	$0.190775\pi$
$12$	0	0
$13$	0.899642	0.249516	0.124758	−	0.992187i	$-0.460185\pi$
$13$	0.899642	0.249516	0.124758	+	0.992187i	$0.460185\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.714260	−0.173234	−0.0866168	−	0.996242i	$-0.527606\pi$
$17$	−0.714260	−0.173234	−0.0866168	+	0.996242i	$0.527606\pi$
$18$	0	0
$19$	−5.35412	−1.22832	−0.614160	−	0.789182i	$-0.710505\pi$
$19$	−5.35412	−1.22832	−0.614160	+	0.789182i	$0.710505\pi$
$20$	0	0
$21$	10.9478	2.38900
$22$	0	0
$23$	3.04153	0.634203	0.317101	−	0.948392i	$-0.397290\pi$
$23$	3.04153	0.634203	0.317101	+	0.948392i	$0.397290\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.55671	0.299589
$28$	0	0
$29$	−0.636866	−0.118263	−0.0591315	−	0.998250i	$-0.518833\pi$
$29$	−0.636866	−0.118263	−0.0591315	+	0.998250i	$0.518833\pi$
$30$	0	0
$31$	−6.03862	−1.08457	−0.542284	−	0.840195i	$-0.682440\pi$
$31$	−6.03862	−1.08457	−0.542284	+	0.840195i	$0.682440\pi$
$32$	0	0
$33$	−12.6395	−2.20026
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.69746	−0.279060	−0.139530	−	0.990218i	$-0.544559\pi$
$37$	−1.69746	−0.279060	−0.139530	+	0.990218i	$0.544559\pi$
$38$	0	0
$39$	−2.07609	−0.332441
$40$	0	0
$41$	10.3516	1.61665	0.808324	−	0.588738i	$-0.200375\pi$
$41$	10.3516	1.61665	0.808324	+	0.588738i	$0.200375\pi$
$42$	0	0
$43$	−8.11200	−1.23707	−0.618534	−	0.785758i	$-0.712273\pi$
$43$	−8.11200	−1.23707	−0.618534	+	0.785758i	$0.712273\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.909166	−0.132616	−0.0663078	−	0.997799i	$-0.521122\pi$
$47$	−0.909166	−0.132616	−0.0663078	+	0.997799i	$0.521122\pi$
$48$	0	0
$49$	15.5059	2.21512
$50$	0	0
$51$	1.64829	0.230807
$52$	0	0
$53$	−1.02940	−0.141399	−0.0706993	−	0.997498i	$-0.522523\pi$
$53$	−1.02940	−0.141399	−0.0706993	+	0.997498i	$0.522523\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.3556	1.63654
$58$	0	0
$59$	−5.59978	−0.729029	−0.364514	−	0.931198i	$-0.618765\pi$
$59$	−5.59978	−0.729029	−0.364514	+	0.931198i	$0.618765\pi$
$60$	0	0
$61$	8.18481	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$61$	8.18481	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$62$	0	0
$63$	−11.0319	−1.38989
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.36315	−0.166535	−0.0832674	−	0.996527i	$-0.526536\pi$
$67$	−1.36315	−0.166535	−0.0832674	+	0.996527i	$0.526536\pi$
$68$	0	0
$69$	−7.01890	−0.844976
$70$	0	0
$71$	−1.56534	−0.185772	−0.0928860	−	0.995677i	$-0.529609\pi$
$71$	−1.56534	−0.185772	−0.0928860	+	0.995677i	$0.529609\pi$
$72$	0	0
$73$	8.11080	0.949297	0.474649	−	0.880175i	$-0.342575\pi$
$73$	8.11080	0.949297	0.474649	+	0.880175i	$0.342575\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−25.9837	−2.96112
$78$	0	0
$79$	12.1629	1.36843	0.684215	−	0.729280i	$-0.260145\pi$
$79$	12.1629	1.36843	0.684215	+	0.729280i	$0.260145\pi$
$80$	0	0
$81$	−10.5687	−1.17430
$82$	0	0
$83$	−12.8315	−1.40844	−0.704218	−	0.709983i	$-0.748702\pi$
$83$	−12.8315	−1.40844	−0.704218	+	0.709983i	$0.748702\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.46969	0.157567
$88$	0	0
$89$	−3.98523	−0.422434	−0.211217	−	0.977439i	$-0.567743\pi$
$89$	−3.98523	−0.422434	−0.211217	+	0.977439i	$0.567743\pi$
$90$	0	0
$91$	−4.26793	−0.447401
$92$	0	0
$93$	13.9353	1.44502
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.95131	−0.908868	−0.454434	−	0.890780i	$-0.650158\pi$
$97$	−8.95131	−0.908868	−0.454434	+	0.890780i	$0.650158\pi$
$98$	0	0
$99$	12.7367	1.28008
$100$	0	0
$101$	4.17849	0.415775	0.207888	−	0.978153i	$-0.433341\pi$
$101$	4.17849	0.415775	0.207888	+	0.978153i	$0.433341\pi$
$102$	0	0
$103$	−9.49112	−0.935187	−0.467594	−	0.883943i	$-0.654879\pi$
$103$	−9.49112	−0.935187	−0.467594	+	0.883943i	$0.654879\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.22384	−0.505008	−0.252504	−	0.967596i	$-0.581254\pi$
$107$	−5.22384	−0.505008	−0.252504	+	0.967596i	$0.581254\pi$
$108$	0	0
$109$	−1.92200	−0.184094	−0.0920469	−	0.995755i	$-0.529341\pi$
$109$	−1.92200	−0.184094	−0.0920469	+	0.995755i	$0.529341\pi$
$110$	0	0
$111$	3.91720	0.371805
$112$	0	0
$113$	2.96549	0.278970	0.139485	−	0.990224i	$-0.455455\pi$
$113$	2.96549	0.278970	0.139485	+	0.990224i	$0.455455\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.09205	0.193410
$118$	0	0
$119$	3.38848	0.310621
$120$	0	0
$121$	18.9991	1.72719
$122$	0	0
$123$	−23.8883	−2.15393
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.9782	−1.15162	−0.575812	−	0.817582i	$-0.695314\pi$
$127$	−12.9782	−1.15162	−0.575812	+	0.817582i	$0.695314\pi$
$128$	0	0
$129$	18.7200	1.64820
$130$	0	0
$131$	2.32841	0.203434	0.101717	−	0.994813i	$-0.467566\pi$
$131$	2.32841	0.203434	0.101717	+	0.994813i	$0.467566\pi$
$132$	0	0
$133$	25.4001	2.20247
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.2552	−1.13246	−0.566232	−	0.824246i	$-0.691600\pi$
$137$	−13.2552	−1.13246	−0.566232	+	0.824246i	$0.691600\pi$
$138$	0	0
$139$	3.85842	0.327267	0.163633	−	0.986521i	$-0.447679\pi$
$139$	3.85842	0.327267	0.163633	+	0.986521i	$0.447679\pi$
$140$	0	0
$141$	2.09807	0.176690
$142$	0	0
$143$	4.92746	0.412055
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−35.7827	−2.95131
$148$	0	0
$149$	−21.2606	−1.74173	−0.870866	−	0.491520i	$-0.836442\pi$
$149$	−21.2606	−1.74173	−0.870866	+	0.491520i	$0.836442\pi$
$150$	0	0
$151$	−13.6617	−1.11177	−0.555886	−	0.831258i	$-0.687621\pi$
$151$	−13.6617	−1.11177	−0.555886	+	0.831258i	$0.687621\pi$
$152$	0	0
$153$	−1.66096	−0.134280
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.4154	−0.831241	−0.415621	−	0.909538i	$-0.636436\pi$
$157$	−10.4154	−0.831241	−0.415621	+	0.909538i	$0.636436\pi$
$158$	0	0
$159$	2.37553	0.188392
$160$	0	0
$161$	−14.4291	−1.13717
$162$	0	0
$163$	8.80154	0.689390	0.344695	−	0.938715i	$-0.387982\pi$
$163$	8.80154	0.689390	0.344695	+	0.938715i	$0.387982\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.3342	1.18659	0.593296	−	0.804984i	$-0.297826\pi$
$167$	15.3342	1.18659	0.593296	+	0.804984i	$0.297826\pi$
$168$	0	0
$169$	−12.1906	−0.937742
$170$	0	0
$171$	−12.4506	−0.952121
$172$	0	0
$173$	−0.436415	−0.0331800	−0.0165900	−	0.999862i	$-0.505281\pi$
$173$	−0.436415	−0.0331800	−0.0165900	+	0.999862i	$0.505281\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.9225	0.971318
$178$	0	0
$179$	3.15305	0.235670	0.117835	−	0.993033i	$-0.462405\pi$
$179$	3.15305	0.235670	0.117835	+	0.993033i	$0.462405\pi$
$180$	0	0
$181$	−2.83593	−0.210793	−0.105397	−	0.994430i	$-0.533611\pi$
$181$	−2.83593	−0.210793	−0.105397	+	0.994430i	$0.533611\pi$
$182$	0	0
$183$	−18.8880	−1.39624
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.91210	−0.286081
$188$	0	0
$189$	−7.38509	−0.537186
$190$	0	0
$191$	−12.8358	−0.928765	−0.464382	−	0.885635i	$-0.653724\pi$
$191$	−12.8358	−0.928765	−0.464382	+	0.885635i	$0.653724\pi$
$192$	0	0
$193$	−12.6967	−0.913931	−0.456965	−	0.889484i	$-0.651064\pi$
$193$	−12.6967	−0.913931	−0.456965	+	0.889484i	$0.651064\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.8846	−1.77296	−0.886479	−	0.462770i	$-0.846856\pi$
$197$	−24.8846	−1.77296	−0.886479	+	0.462770i	$0.846856\pi$
$198$	0	0
$199$	9.23973	0.654987	0.327493	−	0.944853i	$-0.393796\pi$
$199$	9.23973	0.654987	0.327493	+	0.944853i	$0.393796\pi$
$200$	0	0
$201$	3.14572	0.221882
$202$	0	0
$203$	3.02131	0.212055
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.07285	0.491597
$208$	0	0
$209$	−29.3253	−2.02847
$210$	0	0
$211$	11.8417	0.815214	0.407607	−	0.913158i	$-0.366363\pi$
$211$	11.8417	0.815214	0.407607	+	0.913158i	$0.366363\pi$
$212$	0	0
$213$	3.61232	0.247512
$214$	0	0
$215$	0	0
$216$	0	0
$217$	28.6474	1.94471
$218$	0	0
$219$	−18.7172	−1.26479
$220$	0	0
$221$	−0.642578	−0.0432245
$222$	0	0
$223$	17.4171	1.16634	0.583168	−	0.812352i	$-0.301813\pi$
$223$	17.4171	1.16634	0.583168	+	0.812352i	$0.301813\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.2768	0.814838	0.407419	−	0.913241i	$-0.366429\pi$
$227$	12.2768	0.814838	0.407419	+	0.913241i	$0.366429\pi$
$228$	0	0
$229$	−3.59181	−0.237354	−0.118677	−	0.992933i	$-0.537865\pi$
$229$	−3.59181	−0.237354	−0.118677	+	0.992933i	$0.537865\pi$
$230$	0	0
$231$	59.9624	3.94523
$232$	0	0
$233$	11.5258	0.755078	0.377539	−	0.925994i	$-0.376770\pi$
$233$	11.5258	0.755078	0.377539	+	0.925994i	$0.376770\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−28.0681	−1.82322
$238$	0	0
$239$	−26.1479	−1.69137	−0.845684	−	0.533684i	$-0.820807\pi$
$239$	−26.1479	−1.69137	−0.845684	+	0.533684i	$0.820807\pi$
$240$	0	0
$241$	13.8359	0.891248	0.445624	−	0.895220i	$-0.352982\pi$
$241$	13.8359	0.891248	0.445624	+	0.895220i	$0.352982\pi$
$242$	0	0
$243$	19.7191	1.26498
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.81679	−0.306485
$248$	0	0
$249$	29.6110	1.87652
$250$	0	0
$251$	21.1529	1.33516	0.667581	−	0.744537i	$-0.267330\pi$
$251$	21.1529	1.33516	0.667581	+	0.744537i	$0.267330\pi$
$252$	0	0
$253$	16.6589	1.04733
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.6839	−0.666442	−0.333221	−	0.942849i	$-0.608135\pi$
$257$	−10.6839	−0.666442	−0.333221	+	0.942849i	$0.608135\pi$
$258$	0	0
$259$	8.05280	0.500377
$260$	0	0
$261$	−1.48098	−0.0916706
$262$	0	0
$263$	0.0797689	0.00491876	0.00245938	−	0.999997i	$-0.499217\pi$
$263$	0.0797689	0.00491876	0.00245938	+	0.999997i	$0.499217\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.19667	0.562827
$268$	0	0
$269$	3.61927	0.220670	0.110335	−	0.993894i	$-0.464808\pi$
$269$	3.61927	0.220670	0.110335	+	0.993894i	$0.464808\pi$
$270$	0	0
$271$	−10.6923	−0.649514	−0.324757	−	0.945798i	$-0.605282\pi$
$271$	−10.6923	−0.649514	−0.324757	+	0.945798i	$0.605282\pi$
$272$	0	0
$273$	9.84905	0.596092
$274$	0	0
$275$	0	0
$276$	0	0
$277$	32.6994	1.96472	0.982360	−	0.187001i	$-0.0598768\pi$
$277$	32.6994	1.96472	0.982360	+	0.187001i	$0.0598768\pi$
$278$	0	0
$279$	−14.0424	−0.840694
$280$	0	0
$281$	−16.2181	−0.967489	−0.483745	−	0.875209i	$-0.660724\pi$
$281$	−16.2181	−0.967489	−0.483745	+	0.875209i	$0.660724\pi$
$282$	0	0
$283$	32.1766	1.91270	0.956352	−	0.292218i	$-0.0943934\pi$
$283$	32.1766	1.91270	0.956352	+	0.292218i	$0.0943934\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−49.1083	−2.89877
$288$	0	0
$289$	−16.4898	−0.969990
$290$	0	0
$291$	20.6568	1.21093
$292$	0	0
$293$	24.6970	1.44281	0.721406	−	0.692513i	$-0.243496\pi$
$293$	24.6970	1.44281	0.721406	+	0.692513i	$0.243496\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.52632	0.494747
$298$	0	0
$299$	2.73629	0.158244
$300$	0	0
$301$	38.4836	2.21816
$302$	0	0
$303$	−9.64265	−0.553956
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.1625	1.83561	0.917806	−	0.397029i	$-0.129959\pi$
$307$	32.1625	1.83561	0.917806	+	0.397029i	$0.129959\pi$
$308$	0	0
$309$	21.9025	1.24599
$310$	0	0
$311$	21.9578	1.24511	0.622556	−	0.782575i	$-0.286094\pi$
$311$	21.9578	1.24511	0.622556	+	0.782575i	$0.286094\pi$
$312$	0	0
$313$	5.45024	0.308066	0.154033	−	0.988066i	$-0.450774\pi$
$313$	5.45024	0.308066	0.154033	+	0.988066i	$0.450774\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.142772	−0.00801889	−0.00400944	−	0.999992i	$-0.501276\pi$
$317$	−0.142772	−0.00801889	−0.00400944	+	0.999992i	$0.501276\pi$
$318$	0	0
$319$	−3.48820	−0.195302
$320$	0	0
$321$	12.0550	0.672844
$322$	0	0
$323$	3.82423	0.212786
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.43537	0.245276
$328$	0	0
$329$	4.31312	0.237790
$330$	0	0
$331$	23.3536	1.28363	0.641815	−	0.766859i	$-0.278182\pi$
$331$	23.3536	1.28363	0.641815	+	0.766859i	$0.278182\pi$
$332$	0	0
$333$	−3.94731	−0.216311
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.2545	0.885438	0.442719	−	0.896660i	$-0.354014\pi$
$337$	16.2545	0.885438	0.442719	+	0.896660i	$0.354014\pi$
$338$	0	0
$339$	−6.84344	−0.371684
$340$	0	0
$341$	−33.0744	−1.79108
$342$	0	0
$343$	−40.3522	−2.17881
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.87801	0.369231	0.184616	−	0.982811i	$-0.440896\pi$
$347$	6.87801	0.369231	0.184616	+	0.982811i	$0.440896\pi$
$348$	0	0
$349$	−14.7392	−0.788970	−0.394485	−	0.918902i	$-0.629077\pi$
$349$	−14.7392	−0.788970	−0.394485	+	0.918902i	$0.629077\pi$
$350$	0	0
$351$	1.40048	0.0747522
$352$	0	0
$353$	25.0772	1.33473	0.667363	−	0.744733i	$-0.267423\pi$
$353$	25.0772	1.33473	0.667363	+	0.744733i	$0.267423\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.81954	−0.413854
$358$	0	0
$359$	−3.04345	−0.160627	−0.0803135	−	0.996770i	$-0.525592\pi$
$359$	−3.04345	−0.160627	−0.0803135	+	0.996770i	$0.525592\pi$
$360$	0	0
$361$	9.66660	0.508769
$362$	0	0
$363$	−43.8439	−2.30121
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.20929	−0.219723	−0.109862	−	0.993947i	$-0.535041\pi$
$367$	−4.20929	−0.219723	−0.109862	+	0.993947i	$0.535041\pi$
$368$	0	0
$369$	24.0719	1.25313
$370$	0	0
$371$	4.88350	0.253538
$372$	0	0
$373$	17.9138	0.927540	0.463770	−	0.885956i	$-0.346496\pi$
$373$	17.9138	0.927540	0.463770	+	0.885956i	$0.346496\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.572951	−0.0295085
$378$	0	0
$379$	14.8367	0.762110	0.381055	−	0.924552i	$-0.375561\pi$
$379$	14.8367	0.762110	0.381055	+	0.924552i	$0.375561\pi$
$380$	0	0
$381$	29.9495	1.53436
$382$	0	0
$383$	−10.8176	−0.552754	−0.276377	−	0.961049i	$-0.589134\pi$
$383$	−10.8176	−0.552754	−0.276377	+	0.961049i	$0.589134\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.8639	−0.958903
$388$	0	0
$389$	23.0350	1.16792	0.583961	−	0.811782i	$-0.301502\pi$
$389$	23.0350	1.16792	0.583961	+	0.811782i	$0.301502\pi$
$390$	0	0
$391$	−2.17244	−0.109865
$392$	0	0
$393$	−5.37323	−0.271044
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.73700	−0.187555	−0.0937774	−	0.995593i	$-0.529894\pi$
$397$	−3.73700	−0.187555	−0.0937774	+	0.995593i	$0.529894\pi$
$398$	0	0
$399$	−58.6156	−2.93445
$400$	0	0
$401$	−21.9778	−1.09752	−0.548760	−	0.835980i	$-0.684900\pi$
$401$	−21.9778	−1.09752	−0.548760	+	0.835980i	$0.684900\pi$
$402$	0	0
$403$	−5.43260	−0.270617
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.29722	−0.460846
$408$	0	0
$409$	29.2810	1.44785	0.723925	−	0.689878i	$-0.242336\pi$
$409$	29.2810	1.44785	0.723925	+	0.689878i	$0.242336\pi$
$410$	0	0
$411$	30.5888	1.50883
$412$	0	0
$413$	26.5655	1.30721
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.90403	−0.436032
$418$	0	0
$419$	40.6695	1.98684	0.993418	−	0.114546i	$-0.0365414\pi$
$419$	40.6695	1.98684	0.993418	+	0.114546i	$0.0365414\pi$
$420$	0	0
$421$	−28.2203	−1.37538	−0.687688	−	0.726007i	$-0.741374\pi$
$421$	−28.2203	−1.37538	−0.687688	+	0.726007i	$0.741374\pi$
$422$	0	0
$423$	−2.11420	−0.102796
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−38.8290	−1.87907
$428$	0	0
$429$	−11.3710	−0.548999
$430$	0	0
$431$	−3.00603	−0.144795	−0.0723977	−	0.997376i	$-0.523065\pi$
$431$	−3.00603	−0.144795	−0.0723977	+	0.997376i	$0.523065\pi$
$432$	0	0
$433$	4.43785	0.213269	0.106635	−	0.994298i	$-0.465992\pi$
$433$	4.43785	0.213269	0.106635	+	0.994298i	$0.465992\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.2847	−0.779003
$438$	0	0
$439$	−18.3759	−0.877033	−0.438517	−	0.898723i	$-0.644496\pi$
$439$	−18.3759	−0.877033	−0.438517	+	0.898723i	$0.644496\pi$
$440$	0	0
$441$	36.0577	1.71703
$442$	0	0
$443$	−8.18925	−0.389083	−0.194541	−	0.980894i	$-0.562322\pi$
$443$	−8.18925	−0.389083	−0.194541	+	0.980894i	$0.562322\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	49.0627	2.32059
$448$	0	0
$449$	−21.4953	−1.01442	−0.507212	−	0.861821i	$-0.669324\pi$
$449$	−21.4953	−1.01442	−0.507212	+	0.861821i	$0.669324\pi$
$450$	0	0
$451$	56.6971	2.66976
$452$	0	0
$453$	31.5269	1.48126
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.9426	0.932875	0.466438	−	0.884554i	$-0.345537\pi$
$457$	19.9426	0.932875	0.466438	+	0.884554i	$0.345537\pi$
$458$	0	0
$459$	−1.11190	−0.0518989
$460$	0	0
$461$	−24.6510	−1.14811	−0.574056	−	0.818816i	$-0.694631\pi$
$461$	−24.6510	−1.14811	−0.574056	+	0.818816i	$0.694631\pi$
$462$	0	0
$463$	31.9135	1.48314	0.741572	−	0.670873i	$-0.234080\pi$
$463$	31.9135	1.48314	0.741572	+	0.670873i	$0.234080\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.77071	0.267037	0.133518	−	0.991046i	$-0.457373\pi$
$467$	5.77071	0.267037	0.133518	+	0.991046i	$0.457373\pi$
$468$	0	0
$469$	6.46682	0.298610
$470$	0	0
$471$	24.0355	1.10750
$472$	0	0
$473$	−44.4306	−2.04292
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.39378	−0.109604
$478$	0	0
$479$	−37.0122	−1.69113	−0.845564	−	0.533873i	$-0.820736\pi$
$479$	−37.0122	−1.69113	−0.845564	+	0.533873i	$0.820736\pi$
$480$	0	0
$481$	−1.52710	−0.0696299
$482$	0	0
$483$	33.2979	1.51511
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.77445	−0.0804079	−0.0402039	−	0.999191i	$-0.512801\pi$
$487$	−1.77445	−0.0804079	−0.0402039	+	0.999191i	$0.512801\pi$
$488$	0	0
$489$	−20.3112	−0.918505
$490$	0	0
$491$	−6.78582	−0.306240	−0.153120	−	0.988208i	$-0.548932\pi$
$491$	−6.78582	−0.306240	−0.153120	+	0.988208i	$0.548932\pi$
$492$	0	0
$493$	0.454888	0.0204871
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.42604	0.333104
$498$	0	0
$499$	31.3417	1.40305	0.701524	−	0.712645i	$-0.252503\pi$
$499$	31.3417	1.40305	0.701524	+	0.712645i	$0.252503\pi$
$500$	0	0
$501$	−35.3864	−1.58095
$502$	0	0
$503$	27.0025	1.20398	0.601991	−	0.798503i	$-0.294374\pi$
$503$	27.0025	1.20398	0.601991	+	0.798503i	$0.294374\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.1322	1.24940
$508$	0	0
$509$	8.28043	0.367024	0.183512	−	0.983018i	$-0.441253\pi$
$509$	8.28043	0.367024	0.183512	+	0.983018i	$0.441253\pi$
$510$	0	0
$511$	−38.4779	−1.70216
$512$	0	0
$513$	−8.33481	−0.367991
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.97963	−0.219004
$518$	0	0
$519$	1.00711	0.0442072
$520$	0	0
$521$	31.9242	1.39863	0.699313	−	0.714815i	$-0.253489\pi$
$521$	31.9242	1.39863	0.699313	+	0.714815i	$0.253489\pi$
$522$	0	0
$523$	−42.5125	−1.85894	−0.929471	−	0.368895i	$-0.879736\pi$
$523$	−42.5125	−1.85894	−0.929471	+	0.368895i	$0.879736\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.31315	0.187884
$528$	0	0
$529$	−13.7491	−0.597787
$530$	0	0
$531$	−13.0219	−0.565101
$532$	0	0
$533$	9.31273	0.403379
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.27625	−0.313993
$538$	0	0
$539$	84.9278	3.65810
$540$	0	0
$541$	20.1263	0.865295	0.432648	−	0.901563i	$-0.357579\pi$
$541$	20.1263	0.865295	0.432648	+	0.901563i	$0.357579\pi$
$542$	0	0
$543$	6.54444	0.280849
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.86581	0.208047	0.104023	−	0.994575i	$-0.466828\pi$
$547$	4.86581	0.208047	0.104023	+	0.994575i	$0.466828\pi$
$548$	0	0
$549$	19.0332	0.812316
$550$	0	0
$551$	3.40986	0.145265
$552$	0	0
$553$	−57.7011	−2.45370
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.9342	0.971754	0.485877	−	0.874027i	$-0.338500\pi$
$557$	22.9342	0.971754	0.485877	+	0.874027i	$0.338500\pi$
$558$	0	0
$559$	−7.29790	−0.308668
$560$	0	0
$561$	9.02791	0.381159
$562$	0	0
$563$	16.8445	0.709913	0.354956	−	0.934883i	$-0.384496\pi$
$563$	16.8445	0.709913	0.354956	+	0.934883i	$0.384496\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	50.1382	2.10561
$568$	0	0
$569$	24.8021	1.03976	0.519878	−	0.854240i	$-0.325977\pi$
$569$	24.8021	1.03976	0.519878	+	0.854240i	$0.325977\pi$
$570$	0	0
$571$	−26.9990	−1.12987	−0.564936	−	0.825135i	$-0.691099\pi$
$571$	−26.9990	−1.12987	−0.564936	+	0.825135i	$0.691099\pi$
$572$	0	0
$573$	29.6210	1.23743
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.3393	0.513690	0.256845	−	0.966453i	$-0.417317\pi$
$577$	12.3393	0.513690	0.256845	+	0.966453i	$0.417317\pi$
$578$	0	0
$579$	29.3001	1.21767
$580$	0	0
$581$	60.8730	2.52544
$582$	0	0
$583$	−5.63815	−0.233508
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.24878	−0.175366	−0.0876830	−	0.996148i	$-0.527946\pi$
$587$	−4.24878	−0.175366	−0.0876830	+	0.996148i	$0.527946\pi$
$588$	0	0
$589$	32.3315	1.33220
$590$	0	0
$591$	57.4260	2.36219
$592$	0	0
$593$	4.87578	0.200224	0.100112	−	0.994976i	$-0.468080\pi$
$593$	4.87578	0.200224	0.100112	+	0.994976i	$0.468080\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−21.3224	−0.872668
$598$	0	0
$599$	−3.33018	−0.136068	−0.0680338	−	0.997683i	$-0.521673\pi$
$599$	−3.33018	−0.136068	−0.0680338	+	0.997683i	$0.521673\pi$
$600$	0	0
$601$	21.2834	0.868169	0.434084	−	0.900872i	$-0.357072\pi$
$601$	21.2834	0.868169	0.434084	+	0.900872i	$0.357072\pi$
$602$	0	0
$603$	−3.16989	−0.129088
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.70241	0.231454	0.115727	−	0.993281i	$-0.463080\pi$
$607$	5.70241	0.231454	0.115727	+	0.993281i	$0.463080\pi$
$608$	0	0
$609$	−6.97225	−0.282530
$610$	0	0
$611$	−0.817924	−0.0330896
$612$	0	0
$613$	−14.9643	−0.604402	−0.302201	−	0.953244i	$-0.597721\pi$
$613$	−14.9643	−0.604402	−0.302201	+	0.953244i	$0.597721\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.6535	0.589929	0.294965	−	0.955508i	$-0.404692\pi$
$617$	14.6535	0.589929	0.294965	+	0.955508i	$0.404692\pi$
$618$	0	0
$619$	−1.22892	−0.0493946	−0.0246973	−	0.999695i	$-0.507862\pi$
$619$	−1.22892	−0.0493946	−0.0246973	+	0.999695i	$0.507862\pi$
$620$	0	0
$621$	4.73478	0.190000
$622$	0	0
$623$	18.9061	0.757456
$624$	0	0
$625$	0	0
$626$	0	0
$627$	67.6736	2.70262
$628$	0	0
$629$	1.21243	0.0483426
$630$	0	0
$631$	7.36383	0.293149	0.146575	−	0.989200i	$-0.453175\pi$
$631$	7.36383	0.293149	0.146575	+	0.989200i	$0.453175\pi$
$632$	0	0
$633$	−27.3269	−1.08615
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.9497	0.552708
$638$	0	0
$639$	−3.64009	−0.144000
$640$	0	0
$641$	29.2520	1.15538	0.577692	−	0.816255i	$-0.303953\pi$
$641$	29.2520	1.15538	0.577692	+	0.816255i	$0.303953\pi$
$642$	0	0
$643$	−4.03921	−0.159291	−0.0796454	−	0.996823i	$-0.525379\pi$
$643$	−4.03921	−0.159291	−0.0796454	+	0.996823i	$0.525379\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.3993	−0.684036	−0.342018	−	0.939693i	$-0.611110\pi$
$647$	−17.3993	−0.684036	−0.342018	+	0.939693i	$0.611110\pi$
$648$	0	0
$649$	−30.6708	−1.20393
$650$	0	0
$651$	−66.1094	−2.59103
$652$	0	0
$653$	−25.1853	−0.985575	−0.492788	−	0.870150i	$-0.664022\pi$
$653$	−25.1853	−0.985575	−0.492788	+	0.870150i	$0.664022\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.8611	0.735840
$658$	0	0
$659$	20.7512	0.808354	0.404177	−	0.914681i	$-0.367558\pi$
$659$	20.7512	0.808354	0.404177	+	0.914681i	$0.367558\pi$
$660$	0	0
$661$	−25.6646	−0.998235	−0.499118	−	0.866534i	$-0.666342\pi$
$661$	−25.6646	−0.998235	−0.499118	+	0.866534i	$0.666342\pi$
$662$	0	0
$663$	1.48287	0.0575899
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.93705	−0.0750027
$668$	0	0
$669$	−40.1932	−1.55396
$670$	0	0
$671$	44.8294	1.73062
$672$	0	0
$673$	−26.0157	−1.00283	−0.501415	−	0.865207i	$-0.667187\pi$
$673$	−26.0157	−1.00283	−0.501415	+	0.865207i	$0.667187\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.2500	0.470805	0.235403	−	0.971898i	$-0.424359\pi$
$677$	12.2500	0.470805	0.235403	+	0.971898i	$0.424359\pi$
$678$	0	0
$679$	42.4653	1.62967
$680$	0	0
$681$	−28.3309	−1.08564
$682$	0	0
$683$	−32.8161	−1.25567	−0.627837	−	0.778345i	$-0.716060\pi$
$683$	−32.8161	−1.25567	−0.627837	+	0.778345i	$0.716060\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.28878	0.316237
$688$	0	0
$689$	−0.926088	−0.0352812
$690$	0	0
$691$	41.5653	1.58122	0.790609	−	0.612321i	$-0.209764\pi$
$691$	41.5653	1.58122	0.790609	+	0.612321i	$0.209764\pi$
$692$	0	0
$693$	−60.4232	−2.29529
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.39373	−0.280058
$698$	0	0
$699$	−26.5979	−1.00602
$700$	0	0
$701$	39.2783	1.48352	0.741760	−	0.670666i	$-0.233991\pi$
$701$	39.2783	1.48352	0.741760	+	0.670666i	$0.233991\pi$
$702$	0	0
$703$	9.08840	0.342775
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.8229	−0.745517
$708$	0	0
$709$	−27.7025	−1.04039	−0.520195	−	0.854047i	$-0.674141\pi$
$709$	−27.7025	−1.04039	−0.520195	+	0.854047i	$0.674141\pi$
$710$	0	0
$711$	28.2838	1.06073
$712$	0	0
$713$	−18.3666	−0.687836
$714$	0	0
$715$	0	0
$716$	0	0
$717$	60.3412	2.25348
$718$	0	0
$719$	40.4954	1.51022	0.755112	−	0.655596i	$-0.227583\pi$
$719$	40.4954	1.51022	0.755112	+	0.655596i	$0.227583\pi$
$720$	0	0
$721$	45.0262	1.67686
$722$	0	0
$723$	−31.9289	−1.18745
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.8512	1.10712	0.553559	−	0.832810i	$-0.313269\pi$
$727$	29.8512	1.10712	0.553559	+	0.832810i	$0.313269\pi$
$728$	0	0
$729$	−13.7994	−0.511091
$730$	0	0
$731$	5.79408	0.214302
$732$	0	0
$733$	8.18980	0.302497	0.151249	−	0.988496i	$-0.451671\pi$
$733$	8.18980	0.302497	0.151249	+	0.988496i	$0.451671\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.46615	−0.275019
$738$	0	0
$739$	−17.2341	−0.633968	−0.316984	−	0.948431i	$-0.602670\pi$
$739$	−17.2341	−0.633968	−0.316984	+	0.948431i	$0.602670\pi$
$740$	0	0
$741$	11.1156	0.408343
$742$	0	0
$743$	−33.0772	−1.21348	−0.606742	−	0.794899i	$-0.707524\pi$
$743$	−33.0772	−1.21348	−0.606742	+	0.794899i	$0.707524\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−29.8386	−1.09174
$748$	0	0
$749$	24.7821	0.905517
$750$	0	0
$751$	8.95863	0.326905	0.163453	−	0.986551i	$-0.447737\pi$
$751$	8.95863	0.326905	0.163453	+	0.986551i	$0.447737\pi$
$752$	0	0
$753$	−48.8144	−1.77890
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.5769	1.51114	0.755569	−	0.655069i	$-0.227360\pi$
$757$	41.5769	1.51114	0.755569	+	0.655069i	$0.227360\pi$
$758$	0	0
$759$	−38.4435	−1.39541
$760$	0	0
$761$	28.8406	1.04547	0.522735	−	0.852495i	$-0.324912\pi$
$761$	28.8406	1.04547	0.522735	+	0.852495i	$0.324912\pi$
$762$	0	0
$763$	9.11802	0.330094
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.03779	−0.181904
$768$	0	0
$769$	−17.9616	−0.647711	−0.323855	−	0.946107i	$-0.604979\pi$
$769$	−17.9616	−0.647711	−0.323855	+	0.946107i	$0.604979\pi$
$770$	0	0
$771$	24.6550	0.887930
$772$	0	0
$773$	11.8447	0.426025	0.213013	−	0.977049i	$-0.431672\pi$
$773$	11.8447	0.426025	0.213013	+	0.977049i	$0.431672\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−18.5834	−0.666674
$778$	0	0
$779$	−55.4237	−1.98576
$780$	0	0
$781$	−8.57360	−0.306788
$782$	0	0
$783$	−0.991415	−0.0354303
$784$	0	0
$785$	0	0
$786$	0	0
$787$	27.9502	0.996317	0.498158	−	0.867086i	$-0.334010\pi$
$787$	27.9502	0.996317	0.498158	+	0.867086i	$0.334010\pi$
$788$	0	0
$789$	−0.184082	−0.00655349
$790$	0	0
$791$	−14.0684	−0.500215
$792$	0	0
$793$	7.36340	0.261482
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6096	0.588344	0.294172	−	0.955753i	$-0.404956\pi$
$797$	16.6096	0.588344	0.294172	+	0.955753i	$0.404956\pi$
$798$	0	0
$799$	0.649381	0.0229735
$800$	0	0
$801$	−9.26735	−0.327446
$802$	0	0
$803$	44.4240	1.56769
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.35214	−0.294009
$808$	0	0
$809$	−18.0186	−0.633502	−0.316751	−	0.948509i	$-0.602592\pi$
$809$	−18.0186	−0.633502	−0.316751	+	0.948509i	$0.602592\pi$
$810$	0	0
$811$	−34.6100	−1.21532	−0.607661	−	0.794197i	$-0.707892\pi$
$811$	−34.6100	−1.21532	−0.607661	+	0.794197i	$0.707892\pi$
$812$	0	0
$813$	24.6746	0.865376
$814$	0	0
$815$	0	0
$816$	0	0
$817$	43.4326	1.51952
$818$	0	0
$819$	−9.92475	−0.346799
$820$	0	0
$821$	−7.09977	−0.247784	−0.123892	−	0.992296i	$-0.539538\pi$
$821$	−7.09977	−0.247784	−0.123892	+	0.992296i	$0.539538\pi$
$822$	0	0
$823$	50.3585	1.75539	0.877694	−	0.479222i	$-0.159081\pi$
$823$	50.3585	1.75539	0.877694	+	0.479222i	$0.159081\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.4871	−0.503764	−0.251882	−	0.967758i	$-0.581050\pi$
$827$	−14.4871	−0.503764	−0.251882	+	0.967758i	$0.581050\pi$
$828$	0	0
$829$	−44.5915	−1.54873	−0.774364	−	0.632741i	$-0.781930\pi$
$829$	−44.5915	−1.54873	−0.774364	+	0.632741i	$0.781930\pi$
$830$	0	0
$831$	−75.4601	−2.61768
$832$	0	0
$833$	−11.0752	−0.383734
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.40039	−0.324925
$838$	0	0
$839$	30.7138	1.06036	0.530179	−	0.847886i	$-0.322125\pi$
$839$	30.7138	1.06036	0.530179	+	0.847886i	$0.322125\pi$
$840$	0	0
$841$	−28.5944	−0.986014
$842$	0	0
$843$	37.4263	1.28903
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−90.1322	−3.09698
$848$	0	0
$849$	−74.2537	−2.54838
$850$	0	0
$851$	−5.16287	−0.176981
$852$	0	0
$853$	15.4440	0.528794	0.264397	−	0.964414i	$-0.414827\pi$
$853$	15.4440	0.528794	0.264397	+	0.964414i	$0.414827\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.7161	−1.35668	−0.678338	−	0.734750i	$-0.737299\pi$
$857$	−39.7161	−1.35668	−0.678338	+	0.734750i	$0.737299\pi$
$858$	0	0
$859$	34.2532	1.16870	0.584351	−	0.811501i	$-0.301349\pi$
$859$	34.2532	1.16870	0.584351	+	0.811501i	$0.301349\pi$
$860$	0	0
$861$	113.327	3.86216
$862$	0	0
$863$	4.17351	0.142068	0.0710339	−	0.997474i	$-0.477370\pi$
$863$	4.17351	0.142068	0.0710339	+	0.997474i	$0.477370\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	38.0534	1.29236
$868$	0	0
$869$	66.6178	2.25985
$870$	0	0
$871$	−1.22634	−0.0415531
$872$	0	0
$873$	−20.8156	−0.704501
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.37736	−0.215348	−0.107674	−	0.994186i	$-0.534340\pi$
$877$	−6.37736	−0.215348	−0.107674	+	0.994186i	$0.534340\pi$
$878$	0	0
$879$	−56.9929	−1.92232
$880$	0	0
$881$	−2.59628	−0.0874710	−0.0437355	−	0.999043i	$-0.513926\pi$
$881$	−2.59628	−0.0874710	−0.0437355	+	0.999043i	$0.513926\pi$
$882$	0	0
$883$	−2.71824	−0.0914761	−0.0457380	−	0.998953i	$-0.514564\pi$
$883$	−2.71824	−0.0914761	−0.0457380	+	0.998953i	$0.514564\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.6508	0.861271	0.430635	−	0.902526i	$-0.358290\pi$
$887$	25.6508	0.861271	0.430635	+	0.902526i	$0.358290\pi$
$888$	0	0
$889$	61.5688	2.06495
$890$	0	0
$891$	−57.8861	−1.93926
$892$	0	0
$893$	4.86778	0.162894
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.31449	−0.210835
$898$	0	0
$899$	3.84579	0.128264
$900$	0	0
$901$	0.735257	0.0244950
$902$	0	0
$903$	−88.8082	−2.95535
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.64330	−0.154178	−0.0770892	−	0.997024i	$-0.524563\pi$
$907$	−4.64330	−0.154178	−0.0770892	+	0.997024i	$0.524563\pi$
$908$	0	0
$909$	9.71676	0.322285
$910$	0	0
$911$	−39.4037	−1.30550	−0.652752	−	0.757572i	$-0.726386\pi$
$911$	−39.4037	−1.30550	−0.652752	+	0.757572i	$0.726386\pi$
$912$	0	0
$913$	−70.2798	−2.32592
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.0460	−0.364772
$918$	0	0
$919$	−47.1752	−1.55617	−0.778083	−	0.628162i	$-0.783808\pi$
$919$	−47.1752	−1.55617	−0.778083	+	0.628162i	$0.783808\pi$
$920$	0	0
$921$	−74.2211	−2.44567
$922$	0	0
$923$	−1.40825	−0.0463530
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−22.0709	−0.724903
$928$	0	0
$929$	−15.2497	−0.500328	−0.250164	−	0.968204i	$-0.580484\pi$
$929$	−15.2497	−0.500328	−0.250164	+	0.968204i	$0.580484\pi$
$930$	0	0
$931$	−83.0203	−2.72088
$932$	0	0
$933$	−50.6717	−1.65892
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.8071	−0.843080	−0.421540	−	0.906810i	$-0.638510\pi$
$937$	−25.8071	−0.843080	−0.421540	+	0.906810i	$0.638510\pi$
$938$	0	0
$939$	−12.5774	−0.410449
$940$	0	0
$941$	−25.2451	−0.822965	−0.411483	−	0.911418i	$-0.634989\pi$
$941$	−25.2451	−0.822965	−0.411483	+	0.911418i	$0.634989\pi$
$942$	0	0
$943$	31.4847	1.02528
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.7251	0.770963	0.385482	−	0.922716i	$-0.374035\pi$
$947$	23.7251	0.770963	0.385482	+	0.922716i	$0.374035\pi$
$948$	0	0
$949$	7.29681	0.236865
$950$	0	0
$951$	0.329474	0.0106839
$952$	0	0
$953$	47.7232	1.54591	0.772953	−	0.634463i	$-0.218779\pi$
$953$	47.7232	1.54591	0.772953	+	0.634463i	$0.218779\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.04968	0.260209
$958$	0	0
$959$	62.8830	2.03060
$960$	0	0
$961$	5.46496	0.176289
$962$	0	0
$963$	−12.1476	−0.391452
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.8705	1.21783	0.608917	−	0.793234i	$-0.291604\pi$
$967$	37.8705	1.21783	0.608917	+	0.793234i	$0.291604\pi$
$968$	0	0
$969$	−8.82514	−0.283504
$970$	0	0
$971$	28.5614	0.916579	0.458290	−	0.888803i	$-0.348462\pi$
$971$	28.5614	0.916579	0.458290	+	0.888803i	$0.348462\pi$
$972$	0	0
$973$	−18.3045	−0.586814
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.5094	−0.688147	−0.344074	−	0.938943i	$-0.611807\pi$
$977$	−21.5094	−0.688147	−0.344074	+	0.938943i	$0.611807\pi$
$978$	0	0
$979$	−21.8277	−0.697615
$980$	0	0
$981$	−4.46946	−0.142699
$982$	0	0
$983$	−0.223199	−0.00711894	−0.00355947	−	0.999994i	$-0.501133\pi$
$983$	−0.223199	−0.00711894	−0.00355947	+	0.999994i	$0.501133\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9.95333	−0.316818
$988$	0	0
$989$	−24.6729	−0.784552
$990$	0	0
$991$	−19.2605	−0.611831	−0.305915	−	0.952059i	$-0.598962\pi$
$991$	−19.2605	−0.611831	−0.305915	+	0.952059i	$0.598962\pi$
$992$	0	0
$993$	−53.8928	−1.71024
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.1455	1.46144	0.730722	−	0.682675i	$-0.239184\pi$
$997$	46.1455	1.46144	0.730722	+	0.682675i	$0.239184\pi$
$998$	0	0
$999$	−2.64245	−0.0836034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bq.1.4		16
4.3	odd	2		5000.2.a.r.1.13		16
5.4	even	2		10000.2.a.br.1.13		16
20.19	odd	2		5000.2.a.q.1.4		16
25.12	odd	20		400.2.y.d.369.2		32
25.23	odd	20		400.2.y.d.129.2		32
100.11	odd	10		1000.2.m.d.601.2		32
100.23	even	20		200.2.q.a.129.7	✓	32
100.27	even	20		1000.2.q.c.649.2		32
100.39	odd	10		1000.2.m.e.601.7		32
100.59	odd	10		1000.2.m.e.401.7		32
100.63	even	20		1000.2.q.c.849.2		32
100.87	even	20		200.2.q.a.169.7	yes	32
100.91	odd	10		1000.2.m.d.401.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.q.a.129.7	✓	32	100.23	even	20
200.2.q.a.169.7	yes	32	100.87	even	20
400.2.y.d.129.2		32	25.23	odd	20
400.2.y.d.369.2		32	25.12	odd	20
1000.2.m.d.401.2		32	100.91	odd	10
1000.2.m.d.601.2		32	100.11	odd	10
1000.2.m.e.401.7		32	100.59	odd	10
1000.2.m.e.601.7		32	100.39	odd	10
1000.2.q.c.649.2		32	100.27	even	20
1000.2.q.c.849.2		32	100.63	even	20
5000.2.a.q.1.4		16	20.19	odd	2
5000.2.a.r.1.13		16	4.3	odd	2
10000.2.a.bq.1.4		16	1.1	even	1	trivial
10000.2.a.br.1.13		16	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30769 q^{3} -4.74404 q^{7} +2.32542 q^{9} +5.47714 q^{11} +0.899642 q^{13} -0.714260 q^{17} -5.35412 q^{19} +10.9478 q^{21} +3.04153 q^{23} +1.55671 q^{27} -0.636866 q^{29} -6.03862 q^{31} -12.6395 q^{33} -1.69746 q^{37} -2.07609 q^{39} +10.3516 q^{41} -8.11200 q^{43} -0.909166 q^{47} +15.5059 q^{49} +1.64829 q^{51} -1.02940 q^{53} +12.3556 q^{57} -5.59978 q^{59} +8.18481 q^{61} -11.0319 q^{63} -1.36315 q^{67} -7.01890 q^{69} -1.56534 q^{71} +8.11080 q^{73} -25.9837 q^{77} +12.1629 q^{79} -10.5687 q^{81} -12.8315 q^{83} +1.46969 q^{87} -3.98523 q^{89} -4.26793 q^{91} +13.9353 q^{93} -8.95131 q^{97} +12.7367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + 12 q^{11} + 10 q^{13} + 8 q^{17} + 12 q^{19} + 8 q^{21} - 12 q^{23} - 22 q^{27} + 16 q^{29} + 2 q^{31} + 24 q^{33} + 22 q^{37} + 4 q^{39} + 20 q^{41} - 26 q^{43} - 24 q^{47}+ \cdots + 46 q^{99}+O(q^{100}) \) 16 * q - 4 * q^3 - 8 * q^7 + 20 * q^9 + 12 * q^11 + 10 * q^13 + 8 * q^17 + 12 * q^19 + 8 * q^21 - 12 * q^23 - 22 * q^27 + 16 * q^29 + 2 * q^31 + 24 * q^33 + 22 * q^37 + 4 * q^39 + 20 * q^41 - 26 * q^43 - 24 * q^47 + 30 * q^49 + 30 * q^51 + 16 * q^53 + 24 * q^57 + 40 * q^59 + 22 * q^61 - 52 * q^63 - 50 * q^67 + 22 * q^69 + 4 * q^71 + 32 * q^73 + 30 * q^77 + 8 * q^79 + 44 * q^81 - 30 * q^83 - 48 * q^87 + 28 * q^89 + 32 * q^91 + 40 * q^93 + 36 * q^97 + 46 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more