LMFDB - Embedded newform 8020.2.a.e.1.8

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:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-2.25358 q^{3} -1.00000 q^{5} -0.859031 q^{7} +2.07862 q^{9} +O(q^{10})$	$q-2.25358 q^{3} -1.00000 q^{5} -0.859031 q^{7} +2.07862 q^{9} -2.82830 q^{11} +1.99167 q^{13} +2.25358 q^{15} +5.17928 q^{17} -5.67438 q^{19} +1.93590 q^{21} +3.79218 q^{23} +1.00000 q^{25} +2.07640 q^{27} -6.51875 q^{29} +7.37796 q^{31} +6.37381 q^{33} +0.859031 q^{35} +1.97284 q^{37} -4.48838 q^{39} +0.390241 q^{41} -5.73494 q^{43} -2.07862 q^{45} -0.312978 q^{47} -6.26207 q^{49} -11.6719 q^{51} -7.73676 q^{53} +2.82830 q^{55} +12.7877 q^{57} -8.04840 q^{59} -0.491964 q^{61} -1.78560 q^{63} -1.99167 q^{65} -14.3753 q^{67} -8.54598 q^{69} +6.08813 q^{71} +11.8320 q^{73} -2.25358 q^{75} +2.42960 q^{77} +4.45646 q^{79} -10.9152 q^{81} -7.68761 q^{83} -5.17928 q^{85} +14.6905 q^{87} +4.34372 q^{89} -1.71090 q^{91} -16.6268 q^{93} +5.67438 q^{95} +5.68437 q^{97} -5.87898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * 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
$3$	−2.25358	−1.30111	−0.650553	−	0.759461i	$-0.725463\pi$
$3$	−2.25358	−1.30111	−0.650553	+	0.759461i	$0.725463\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.859031	−0.324683	−0.162342	−	0.986735i	$-0.551905\pi$
$7$	−0.859031	−0.324683	−0.162342	+	0.986735i	$0.551905\pi$
$8$	0	0
$9$	2.07862	0.692875
$10$	0	0
$11$	−2.82830	−0.852765	−0.426383	−	0.904543i	$-0.640212\pi$
$11$	−2.82830	−0.852765	−0.426383	+	0.904543i	$0.640212\pi$
$12$	0	0
$13$	1.99167	0.552389	0.276194	−	0.961102i	$-0.410927\pi$
$13$	1.99167	0.552389	0.276194	+	0.961102i	$0.410927\pi$
$14$	0	0
$15$	2.25358	0.581872
$16$	0	0
$17$	5.17928	1.25616	0.628081	−	0.778148i	$-0.283841\pi$
$17$	5.17928	1.25616	0.628081	+	0.778148i	$0.283841\pi$
$18$	0	0
$19$	−5.67438	−1.30179	−0.650896	−	0.759167i	$-0.725606\pi$
$19$	−5.67438	−1.30179	−0.650896	+	0.759167i	$0.725606\pi$
$20$	0	0
$21$	1.93590	0.422447
$22$	0	0
$23$	3.79218	0.790724	0.395362	−	0.918525i	$-0.370619\pi$
$23$	3.79218	0.790724	0.395362	+	0.918525i	$0.370619\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.07640	0.399602
$28$	0	0
$29$	−6.51875	−1.21050	−0.605251	−	0.796035i	$-0.706927\pi$
$29$	−6.51875	−1.21050	−0.605251	+	0.796035i	$0.706927\pi$
$30$	0	0
$31$	7.37796	1.32512	0.662560	−	0.749009i	$-0.269470\pi$
$31$	7.37796	1.32512	0.662560	+	0.749009i	$0.269470\pi$
$32$	0	0
$33$	6.37381	1.10954
$34$	0	0
$35$	0.859031	0.145203
$36$	0	0
$37$	1.97284	0.324333	0.162167	−	0.986763i	$-0.448152\pi$
$37$	1.97284	0.324333	0.162167	+	0.986763i	$0.448152\pi$
$38$	0	0
$39$	−4.48838	−0.718716
$40$	0	0
$41$	0.390241	0.0609454	0.0304727	−	0.999536i	$-0.490299\pi$
$41$	0.390241	0.0609454	0.0304727	+	0.999536i	$0.490299\pi$
$42$	0	0
$43$	−5.73494	−0.874570	−0.437285	−	0.899323i	$-0.644060\pi$
$43$	−5.73494	−0.874570	−0.437285	+	0.899323i	$0.644060\pi$
$44$	0	0
$45$	−2.07862	−0.309863
$46$	0	0
$47$	−0.312978	−0.0456525	−0.0228263	−	0.999739i	$-0.507266\pi$
$47$	−0.312978	−0.0456525	−0.0228263	+	0.999739i	$0.507266\pi$
$48$	0	0
$49$	−6.26207	−0.894581
$50$	0	0
$51$	−11.6719	−1.63440
$52$	0	0
$53$	−7.73676	−1.06273	−0.531363	−	0.847144i	$-0.678320\pi$
$53$	−7.73676	−1.06273	−0.531363	+	0.847144i	$0.678320\pi$
$54$	0	0
$55$	2.82830	0.381368
$56$	0	0
$57$	12.7877	1.69377
$58$	0	0
$59$	−8.04840	−1.04781	−0.523906	−	0.851776i	$-0.675526\pi$
$59$	−8.04840	−1.04781	−0.523906	+	0.851776i	$0.675526\pi$
$60$	0	0
$61$	−0.491964	−0.0629896	−0.0314948	−	0.999504i	$-0.510027\pi$
$61$	−0.491964	−0.0629896	−0.0314948	+	0.999504i	$0.510027\pi$
$62$	0	0
$63$	−1.78560	−0.224965
$64$	0	0
$65$	−1.99167	−0.247036
$66$	0	0
$67$	−14.3753	−1.75622	−0.878111	−	0.478457i	$-0.841196\pi$
$67$	−14.3753	−1.75622	−0.878111	+	0.478457i	$0.841196\pi$
$68$	0	0
$69$	−8.54598	−1.02882
$70$	0	0
$71$	6.08813	0.722528	0.361264	−	0.932464i	$-0.382345\pi$
$71$	6.08813	0.722528	0.361264	+	0.932464i	$0.382345\pi$
$72$	0	0
$73$	11.8320	1.38483	0.692413	−	0.721501i	$-0.256548\pi$
$73$	11.8320	1.38483	0.692413	+	0.721501i	$0.256548\pi$
$74$	0	0
$75$	−2.25358	−0.260221
$76$	0	0
$77$	2.42960	0.276879
$78$	0	0
$79$	4.45646	0.501391	0.250695	−	0.968066i	$-0.419341\pi$
$79$	4.45646	0.501391	0.250695	+	0.968066i	$0.419341\pi$
$80$	0	0
$81$	−10.9152	−1.21280
$82$	0	0
$83$	−7.68761	−0.843825	−0.421913	−	0.906636i	$-0.638641\pi$
$83$	−7.68761	−0.843825	−0.421913	+	0.906636i	$0.638641\pi$
$84$	0	0
$85$	−5.17928	−0.561772
$86$	0	0
$87$	14.6905	1.57499
$88$	0	0
$89$	4.34372	0.460434	0.230217	−	0.973139i	$-0.426056\pi$
$89$	4.34372	0.460434	0.230217	+	0.973139i	$0.426056\pi$
$90$	0	0
$91$	−1.71090	−0.179351
$92$	0	0
$93$	−16.6268	−1.72412
$94$	0	0
$95$	5.67438	0.582179
$96$	0	0
$97$	5.68437	0.577161	0.288580	−	0.957456i	$-0.406817\pi$
$97$	5.68437	0.577161	0.288580	+	0.957456i	$0.406817\pi$
$98$	0	0
$99$	−5.87898	−0.590859
$100$	0	0
$101$	1.13775	0.113210	0.0566051	−	0.998397i	$-0.481972\pi$
$101$	1.13775	0.113210	0.0566051	+	0.998397i	$0.481972\pi$
$102$	0	0
$103$	−19.7220	−1.94327	−0.971635	−	0.236486i	$-0.924004\pi$
$103$	−19.7220	−1.94327	−0.971635	+	0.236486i	$0.924004\pi$
$104$	0	0
$105$	−1.93590	−0.188924
$106$	0	0
$107$	−8.02956	−0.776247	−0.388123	−	0.921607i	$-0.626877\pi$
$107$	−8.02956	−0.776247	−0.388123	+	0.921607i	$0.626877\pi$
$108$	0	0
$109$	0.467750	0.0448023	0.0224012	−	0.999749i	$-0.492869\pi$
$109$	0.467750	0.0448023	0.0224012	+	0.999749i	$0.492869\pi$
$110$	0	0
$111$	−4.44596	−0.421992
$112$	0	0
$113$	18.0552	1.69849	0.849244	−	0.528000i	$-0.177058\pi$
$113$	18.0552	1.69849	0.849244	+	0.528000i	$0.177058\pi$
$114$	0	0
$115$	−3.79218	−0.353623
$116$	0	0
$117$	4.13992	0.382736
$118$	0	0
$119$	−4.44917	−0.407855
$120$	0	0
$121$	−3.00071	−0.272791
$122$	0	0
$123$	−0.879440	−0.0792964
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.5514	−1.73490	−0.867451	−	0.497523i	$-0.834243\pi$
$127$	−19.5514	−1.73490	−0.867451	+	0.497523i	$0.834243\pi$
$128$	0	0
$129$	12.9241	1.13791
$130$	0	0
$131$	3.98011	0.347744	0.173872	−	0.984768i	$-0.444372\pi$
$131$	3.98011	0.347744	0.173872	+	0.984768i	$0.444372\pi$
$132$	0	0
$133$	4.87447	0.422670
$134$	0	0
$135$	−2.07640	−0.178708
$136$	0	0
$137$	6.11881	0.522765	0.261383	−	0.965235i	$-0.415822\pi$
$137$	6.11881	0.522765	0.261383	+	0.965235i	$0.415822\pi$
$138$	0	0
$139$	13.4813	1.14347	0.571736	−	0.820438i	$-0.306270\pi$
$139$	13.4813	1.14347	0.571736	+	0.820438i	$0.306270\pi$
$140$	0	0
$141$	0.705321	0.0593988
$142$	0	0
$143$	−5.63303	−0.471058
$144$	0	0
$145$	6.51875	0.541353
$146$	0	0
$147$	14.1121	1.16394
$148$	0	0
$149$	9.91403	0.812189	0.406095	−	0.913831i	$-0.366890\pi$
$149$	9.91403	0.812189	0.406095	+	0.913831i	$0.366890\pi$
$150$	0	0
$151$	19.5149	1.58810	0.794050	−	0.607853i	$-0.207969\pi$
$151$	19.5149	1.58810	0.794050	+	0.607853i	$0.207969\pi$
$152$	0	0
$153$	10.7658	0.870362
$154$	0	0
$155$	−7.37796	−0.592612
$156$	0	0
$157$	−14.4561	−1.15372	−0.576861	−	0.816843i	$-0.695722\pi$
$157$	−14.4561	−1.15372	−0.576861	+	0.816843i	$0.695722\pi$
$158$	0	0
$159$	17.4354	1.38272
$160$	0	0
$161$	−3.25760	−0.256735
$162$	0	0
$163$	7.34353	0.575190	0.287595	−	0.957752i	$-0.407144\pi$
$163$	7.34353	0.575190	0.287595	+	0.957752i	$0.407144\pi$
$164$	0	0
$165$	−6.37381	−0.496200
$166$	0	0
$167$	−20.3900	−1.57783	−0.788914	−	0.614503i	$-0.789356\pi$
$167$	−20.3900	−1.57783	−0.788914	+	0.614503i	$0.789356\pi$
$168$	0	0
$169$	−9.03327	−0.694867
$170$	0	0
$171$	−11.7949	−0.901978
$172$	0	0
$173$	7.97251	0.606139	0.303069	−	0.952969i	$-0.401989\pi$
$173$	7.97251	0.606139	0.303069	+	0.952969i	$0.401989\pi$
$174$	0	0
$175$	−0.859031	−0.0649367
$176$	0	0
$177$	18.1377	1.36331
$178$	0	0
$179$	8.83322	0.660226	0.330113	−	0.943941i	$-0.392913\pi$
$179$	8.83322	0.660226	0.330113	+	0.943941i	$0.392913\pi$
$180$	0	0
$181$	13.4236	0.997770	0.498885	−	0.866668i	$-0.333743\pi$
$181$	13.4236	0.997770	0.498885	+	0.866668i	$0.333743\pi$
$182$	0	0
$183$	1.10868	0.0819561
$184$	0	0
$185$	−1.97284	−0.145046
$186$	0	0
$187$	−14.6486	−1.07121
$188$	0	0
$189$	−1.78369	−0.129744
$190$	0	0
$191$	−8.57609	−0.620544	−0.310272	−	0.950648i	$-0.600420\pi$
$191$	−8.57609	−0.620544	−0.310272	+	0.950648i	$0.600420\pi$
$192$	0	0
$193$	8.49192	0.611262	0.305631	−	0.952150i	$-0.401133\pi$
$193$	8.49192	0.611262	0.305631	+	0.952150i	$0.401133\pi$
$194$	0	0
$195$	4.48838	0.321420
$196$	0	0
$197$	−21.4581	−1.52883	−0.764414	−	0.644726i	$-0.776971\pi$
$197$	−21.4581	−1.52883	−0.764414	+	0.644726i	$0.776971\pi$
$198$	0	0
$199$	3.32067	0.235396	0.117698	−	0.993049i	$-0.462448\pi$
$199$	3.32067	0.235396	0.117698	+	0.993049i	$0.462448\pi$
$200$	0	0
$201$	32.3959	2.28503
$202$	0	0
$203$	5.59981	0.393030
$204$	0	0
$205$	−0.390241	−0.0272556
$206$	0	0
$207$	7.88251	0.547873
$208$	0	0
$209$	16.0489	1.11012
$210$	0	0
$211$	−0.757312	−0.0521355	−0.0260678	−	0.999660i	$-0.508299\pi$
$211$	−0.757312	−0.0521355	−0.0260678	+	0.999660i	$0.508299\pi$
$212$	0	0
$213$	−13.7201	−0.940084
$214$	0	0
$215$	5.73494	0.391119
$216$	0	0
$217$	−6.33790	−0.430245
$218$	0	0
$219$	−26.6643	−1.80181
$220$	0	0
$221$	10.3154	0.693889
$222$	0	0
$223$	−5.95639	−0.398870	−0.199435	−	0.979911i	$-0.563911\pi$
$223$	−5.95639	−0.398870	−0.199435	+	0.979911i	$0.563911\pi$
$224$	0	0
$225$	2.07862	0.138575
$226$	0	0
$227$	−4.65733	−0.309118	−0.154559	−	0.987984i	$-0.549396\pi$
$227$	−4.65733	−0.309118	−0.154559	+	0.987984i	$0.549396\pi$
$228$	0	0
$229$	−11.0157	−0.727937	−0.363968	−	0.931411i	$-0.618578\pi$
$229$	−11.0157	−0.727937	−0.363968	+	0.931411i	$0.618578\pi$
$230$	0	0
$231$	−5.47530	−0.360248
$232$	0	0
$233$	−2.82887	−0.185325	−0.0926627	−	0.995698i	$-0.529538\pi$
$233$	−2.82887	−0.185325	−0.0926627	+	0.995698i	$0.529538\pi$
$234$	0	0
$235$	0.312978	0.0204164
$236$	0	0
$237$	−10.0430	−0.652362
$238$	0	0
$239$	−26.7538	−1.73056	−0.865280	−	0.501289i	$-0.832859\pi$
$239$	−26.7538	−1.73056	−0.865280	+	0.501289i	$0.832859\pi$
$240$	0	0
$241$	−3.09932	−0.199645	−0.0998223	−	0.995005i	$-0.531827\pi$
$241$	−3.09932	−0.199645	−0.0998223	+	0.995005i	$0.531827\pi$
$242$	0	0
$243$	18.3691	1.17838
$244$	0	0
$245$	6.26207	0.400069
$246$	0	0
$247$	−11.3015	−0.719095
$248$	0	0
$249$	17.3247	1.09791
$250$	0	0
$251$	4.55363	0.287423	0.143711	−	0.989620i	$-0.454096\pi$
$251$	4.55363	0.287423	0.143711	+	0.989620i	$0.454096\pi$
$252$	0	0
$253$	−10.7254	−0.674302
$254$	0	0
$255$	11.6719	0.730925
$256$	0	0
$257$	15.2931	0.953958	0.476979	−	0.878915i	$-0.341732\pi$
$257$	15.2931	0.953958	0.476979	+	0.878915i	$0.341732\pi$
$258$	0	0
$259$	−1.69473	−0.105306
$260$	0	0
$261$	−13.5500	−0.838726
$262$	0	0
$263$	5.47978	0.337897	0.168949	−	0.985625i	$-0.445963\pi$
$263$	5.47978	0.337897	0.168949	+	0.985625i	$0.445963\pi$
$264$	0	0
$265$	7.73676	0.475266
$266$	0	0
$267$	−9.78893	−0.599073
$268$	0	0
$269$	−2.51410	−0.153287	−0.0766437	−	0.997059i	$-0.524420\pi$
$269$	−2.51410	−0.153287	−0.0766437	+	0.997059i	$0.524420\pi$
$270$	0	0
$271$	17.8122	1.08202	0.541008	−	0.841017i	$-0.318043\pi$
$271$	17.8122	1.08202	0.541008	+	0.841017i	$0.318043\pi$
$272$	0	0
$273$	3.85566	0.233355
$274$	0	0
$275$	−2.82830	−0.170553
$276$	0	0
$277$	−2.28852	−0.137504	−0.0687520	−	0.997634i	$-0.521902\pi$
$277$	−2.28852	−0.137504	−0.0687520	+	0.997634i	$0.521902\pi$
$278$	0	0
$279$	15.3360	0.918142
$280$	0	0
$281$	25.2173	1.50434	0.752169	−	0.658970i	$-0.229008\pi$
$281$	25.2173	1.50434	0.752169	+	0.658970i	$0.229008\pi$
$282$	0	0
$283$	13.2973	0.790440	0.395220	−	0.918586i	$-0.370668\pi$
$283$	13.2973	0.790440	0.395220	+	0.918586i	$0.370668\pi$
$284$	0	0
$285$	−12.7877	−0.757476
$286$	0	0
$287$	−0.335229	−0.0197880
$288$	0	0
$289$	9.82499	0.577941
$290$	0	0
$291$	−12.8102	−0.750947
$292$	0	0
$293$	27.1331	1.58513	0.792565	−	0.609787i	$-0.208745\pi$
$293$	27.1331	1.58513	0.792565	+	0.609787i	$0.208745\pi$
$294$	0	0
$295$	8.04840	0.468596
$296$	0	0
$297$	−5.87267	−0.340767
$298$	0	0
$299$	7.55275	0.436787
$300$	0	0
$301$	4.92649	0.283958
$302$	0	0
$303$	−2.56401	−0.147298
$304$	0	0
$305$	0.491964	0.0281698
$306$	0	0
$307$	−3.46576	−0.197801	−0.0989006	−	0.995097i	$-0.531533\pi$
$307$	−3.46576	−0.197801	−0.0989006	+	0.995097i	$0.531533\pi$
$308$	0	0
$309$	44.4452	2.52840
$310$	0	0
$311$	12.2344	0.693747	0.346873	−	0.937912i	$-0.387243\pi$
$311$	12.2344	0.693747	0.346873	+	0.937912i	$0.387243\pi$
$312$	0	0
$313$	−18.2672	−1.03253	−0.516263	−	0.856430i	$-0.672677\pi$
$313$	−18.2672	−1.03253	−0.516263	+	0.856430i	$0.672677\pi$
$314$	0	0
$315$	1.78560	0.100607
$316$	0	0
$317$	23.8247	1.33813	0.669065	−	0.743204i	$-0.266695\pi$
$317$	23.8247	1.33813	0.669065	+	0.743204i	$0.266695\pi$
$318$	0	0
$319$	18.4370	1.03227
$320$	0	0
$321$	18.0953	1.00998
$322$	0	0
$323$	−29.3892	−1.63526
$324$	0	0
$325$	1.99167	0.110478
$326$	0	0
$327$	−1.05411	−0.0582925
$328$	0	0
$329$	0.268858	0.0148226
$330$	0	0
$331$	15.0757	0.828638	0.414319	−	0.910132i	$-0.364020\pi$
$331$	15.0757	0.828638	0.414319	+	0.910132i	$0.364020\pi$
$332$	0	0
$333$	4.10080	0.224722
$334$	0	0
$335$	14.3753	0.785406
$336$	0	0
$337$	−29.6811	−1.61683	−0.808415	−	0.588613i	$-0.799674\pi$
$337$	−29.6811	−1.61683	−0.808415	+	0.588613i	$0.799674\pi$
$338$	0	0
$339$	−40.6888	−2.20991
$340$	0	0
$341$	−20.8671	−1.13002
$342$	0	0
$343$	11.3925	0.615139
$344$	0	0
$345$	8.54598	0.460100
$346$	0	0
$347$	−0.905694	−0.0486202	−0.0243101	−	0.999704i	$-0.507739\pi$
$347$	−0.905694	−0.0486202	−0.0243101	+	0.999704i	$0.507739\pi$
$348$	0	0
$349$	−1.25012	−0.0669174	−0.0334587	−	0.999440i	$-0.510652\pi$
$349$	−1.25012	−0.0669174	−0.0334587	+	0.999440i	$0.510652\pi$
$350$	0	0
$351$	4.13549	0.220736
$352$	0	0
$353$	−6.82361	−0.363184	−0.181592	−	0.983374i	$-0.558125\pi$
$353$	−6.82361	−0.363184	−0.181592	+	0.983374i	$0.558125\pi$
$354$	0	0
$355$	−6.08813	−0.323124
$356$	0	0
$357$	10.0266	0.530662
$358$	0	0
$359$	2.81363	0.148498	0.0742489	−	0.997240i	$-0.476344\pi$
$359$	2.81363	0.148498	0.0742489	+	0.997240i	$0.476344\pi$
$360$	0	0
$361$	13.1986	0.694661
$362$	0	0
$363$	6.76233	0.354930
$364$	0	0
$365$	−11.8320	−0.619313
$366$	0	0
$367$	20.4071	1.06524	0.532620	−	0.846354i	$-0.321207\pi$
$367$	20.4071	1.06524	0.532620	+	0.846354i	$0.321207\pi$
$368$	0	0
$369$	0.811164	0.0422275
$370$	0	0
$371$	6.64612	0.345049
$372$	0	0
$373$	35.2211	1.82368	0.911839	−	0.410547i	$-0.134662\pi$
$373$	35.2211	1.82368	0.911839	+	0.410547i	$0.134662\pi$
$374$	0	0
$375$	2.25358	0.116374
$376$	0	0
$377$	−12.9832	−0.668668
$378$	0	0
$379$	0.733334	0.0376688	0.0188344	−	0.999823i	$-0.494004\pi$
$379$	0.733334	0.0376688	0.0188344	+	0.999823i	$0.494004\pi$
$380$	0	0
$381$	44.0605	2.25729
$382$	0	0
$383$	−15.3336	−0.783512	−0.391756	−	0.920069i	$-0.628132\pi$
$383$	−15.3336	−0.783512	−0.391756	+	0.920069i	$0.628132\pi$
$384$	0	0
$385$	−2.42960	−0.123824
$386$	0	0
$387$	−11.9208	−0.605967
$388$	0	0
$389$	31.2956	1.58675	0.793376	−	0.608732i	$-0.208322\pi$
$389$	31.2956	1.58675	0.793376	+	0.608732i	$0.208322\pi$
$390$	0	0
$391$	19.6408	0.993277
$392$	0	0
$393$	−8.96949	−0.452451
$394$	0	0
$395$	−4.45646	−0.224229
$396$	0	0
$397$	24.2191	1.21552	0.607761	−	0.794120i	$-0.292068\pi$
$397$	24.2191	1.21552	0.607761	+	0.794120i	$0.292068\pi$
$398$	0	0
$399$	−10.9850	−0.549938
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	14.6944	0.731982
$404$	0	0
$405$	10.9152	0.542380
$406$	0	0
$407$	−5.57979	−0.276580
$408$	0	0
$409$	9.20488	0.455152	0.227576	−	0.973760i	$-0.426920\pi$
$409$	9.20488	0.455152	0.227576	+	0.973760i	$0.426920\pi$
$410$	0	0
$411$	−13.7892	−0.680172
$412$	0	0
$413$	6.91382	0.340207
$414$	0	0
$415$	7.68761	0.377370
$416$	0	0
$417$	−30.3813	−1.48778
$418$	0	0
$419$	32.5943	1.59234	0.796168	−	0.605075i	$-0.206857\pi$
$419$	32.5943	1.59234	0.796168	+	0.605075i	$0.206857\pi$
$420$	0	0
$421$	−1.63911	−0.0798855	−0.0399427	−	0.999202i	$-0.512718\pi$
$421$	−1.63911	−0.0798855	−0.0399427	+	0.999202i	$0.512718\pi$
$422$	0	0
$423$	−0.650564	−0.0316315
$424$	0	0
$425$	5.17928	0.251232
$426$	0	0
$427$	0.422613	0.0204517
$428$	0	0
$429$	12.6945	0.612896
$430$	0	0
$431$	−7.27893	−0.350613	−0.175307	−	0.984514i	$-0.556092\pi$
$431$	−7.27893	−0.350613	−0.175307	+	0.984514i	$0.556092\pi$
$432$	0	0
$433$	27.3718	1.31541	0.657703	−	0.753278i	$-0.271528\pi$
$433$	27.3718	1.31541	0.657703	+	0.753278i	$0.271528\pi$
$434$	0	0
$435$	−14.6905	−0.704357
$436$	0	0
$437$	−21.5183	−1.02936
$438$	0	0
$439$	26.4478	1.26229	0.631143	−	0.775667i	$-0.282586\pi$
$439$	26.4478	1.26229	0.631143	+	0.775667i	$0.282586\pi$
$440$	0	0
$441$	−13.0165	−0.619832
$442$	0	0
$443$	22.3718	1.06292	0.531458	−	0.847085i	$-0.321644\pi$
$443$	22.3718	1.06292	0.531458	+	0.847085i	$0.321644\pi$
$444$	0	0
$445$	−4.34372	−0.205912
$446$	0	0
$447$	−22.3421	−1.05674
$448$	0	0
$449$	7.87319	0.371559	0.185779	−	0.982591i	$-0.440519\pi$
$449$	7.87319	0.371559	0.185779	+	0.982591i	$0.440519\pi$
$450$	0	0
$451$	−1.10372	−0.0519721
$452$	0	0
$453$	−43.9784	−2.06628
$454$	0	0
$455$	1.71090	0.0802084
$456$	0	0
$457$	19.1147	0.894146	0.447073	−	0.894497i	$-0.352466\pi$
$457$	19.1147	0.894146	0.447073	+	0.894497i	$0.352466\pi$
$458$	0	0
$459$	10.7542	0.501965
$460$	0	0
$461$	39.2734	1.82914	0.914572	−	0.404423i	$-0.132528\pi$
$461$	39.2734	1.82914	0.914572	+	0.404423i	$0.132528\pi$
$462$	0	0
$463$	−34.8437	−1.61932	−0.809661	−	0.586898i	$-0.800349\pi$
$463$	−34.8437	−1.61932	−0.809661	+	0.586898i	$0.800349\pi$
$464$	0	0
$465$	16.6268	0.771051
$466$	0	0
$467$	33.4037	1.54574	0.772870	−	0.634564i	$-0.218820\pi$
$467$	33.4037	1.54574	0.772870	+	0.634564i	$0.218820\pi$
$468$	0	0
$469$	12.3488	0.570216
$470$	0	0
$471$	32.5779	1.50111
$472$	0	0
$473$	16.2201	0.745803
$474$	0	0
$475$	−5.67438	−0.260358
$476$	0	0
$477$	−16.0818	−0.736336
$478$	0	0
$479$	−27.6365	−1.26275	−0.631373	−	0.775479i	$-0.717508\pi$
$479$	−27.6365	−1.26275	−0.631373	+	0.775479i	$0.717508\pi$
$480$	0	0
$481$	3.92924	0.179158
$482$	0	0
$483$	7.34126	0.334039
$484$	0	0
$485$	−5.68437	−0.258114
$486$	0	0
$487$	−34.2529	−1.55215	−0.776074	−	0.630642i	$-0.782792\pi$
$487$	−34.2529	−1.55215	−0.776074	+	0.630642i	$0.782792\pi$
$488$	0	0
$489$	−16.5492	−0.748382
$490$	0	0
$491$	−17.4765	−0.788701	−0.394351	−	0.918960i	$-0.629030\pi$
$491$	−17.4765	−0.788701	−0.394351	+	0.918960i	$0.629030\pi$
$492$	0	0
$493$	−33.7625	−1.52059
$494$	0	0
$495$	5.87898	0.264240
$496$	0	0
$497$	−5.22989	−0.234593
$498$	0	0
$499$	11.3715	0.509056	0.254528	−	0.967065i	$-0.418080\pi$
$499$	11.3715	0.509056	0.254528	+	0.967065i	$0.418080\pi$
$500$	0	0
$501$	45.9506	2.05292
$502$	0	0
$503$	−12.9074	−0.575512	−0.287756	−	0.957704i	$-0.592909\pi$
$503$	−12.9074	−0.575512	−0.287756	+	0.957704i	$0.592909\pi$
$504$	0	0
$505$	−1.13775	−0.0506292
$506$	0	0
$507$	20.3572	0.904095
$508$	0	0
$509$	−38.2661	−1.69612	−0.848058	−	0.529903i	$-0.822228\pi$
$509$	−38.2661	−1.69612	−0.848058	+	0.529903i	$0.822228\pi$
$510$	0	0
$511$	−10.1640	−0.449630
$512$	0	0
$513$	−11.7822	−0.520199
$514$	0	0
$515$	19.7220	0.869057
$516$	0	0
$517$	0.885197	0.0389309
$518$	0	0
$519$	−17.9667	−0.788650
$520$	0	0
$521$	−8.59379	−0.376501	−0.188250	−	0.982121i	$-0.560282\pi$
$521$	−8.59379	−0.376501	−0.188250	+	0.982121i	$0.560282\pi$
$522$	0	0
$523$	3.95926	0.173126	0.0865631	−	0.996246i	$-0.472412\pi$
$523$	3.95926	0.173126	0.0865631	+	0.996246i	$0.472412\pi$
$524$	0	0
$525$	1.93590	0.0844894
$526$	0	0
$527$	38.2126	1.66457
$528$	0	0
$529$	−8.61938	−0.374756
$530$	0	0
$531$	−16.7296	−0.726002
$532$	0	0
$533$	0.777230	0.0336656
$534$	0	0
$535$	8.02956	0.347148
$536$	0	0
$537$	−19.9064	−0.859024
$538$	0	0
$539$	17.7110	0.762867
$540$	0	0
$541$	−8.32259	−0.357816	−0.178908	−	0.983866i	$-0.557256\pi$
$541$	−8.32259	−0.357816	−0.178908	+	0.983866i	$0.557256\pi$
$542$	0	0
$543$	−30.2512	−1.29820
$544$	0	0
$545$	−0.467750	−0.0200362
$546$	0	0
$547$	16.8845	0.721930	0.360965	−	0.932579i	$-0.382447\pi$
$547$	16.8845	0.721930	0.360965	+	0.932579i	$0.382447\pi$
$548$	0	0
$549$	−1.02261	−0.0436439
$550$	0	0
$551$	36.9899	1.57582
$552$	0	0
$553$	−3.82824	−0.162793
$554$	0	0
$555$	4.44596	0.188720
$556$	0	0
$557$	18.0211	0.763578	0.381789	−	0.924250i	$-0.375308\pi$
$557$	18.0211	0.763578	0.381789	+	0.924250i	$0.375308\pi$
$558$	0	0
$559$	−11.4221	−0.483102
$560$	0	0
$561$	33.0118	1.39376
$562$	0	0
$563$	−4.37273	−0.184289	−0.0921444	−	0.995746i	$-0.529372\pi$
$563$	−4.37273	−0.184289	−0.0921444	+	0.995746i	$0.529372\pi$
$564$	0	0
$565$	−18.0552	−0.759587
$566$	0	0
$567$	9.37649	0.393776
$568$	0	0
$569$	25.9819	1.08922	0.544608	−	0.838691i	$-0.316678\pi$
$569$	25.9819	1.08922	0.544608	+	0.838691i	$0.316678\pi$
$570$	0	0
$571$	34.4784	1.44288	0.721439	−	0.692478i	$-0.243481\pi$
$571$	34.4784	1.44288	0.721439	+	0.692478i	$0.243481\pi$
$572$	0	0
$573$	19.3269	0.807393
$574$	0	0
$575$	3.79218	0.158145
$576$	0	0
$577$	−4.63057	−0.192773	−0.0963866	−	0.995344i	$-0.530729\pi$
$577$	−4.63057	−0.192773	−0.0963866	+	0.995344i	$0.530729\pi$
$578$	0	0
$579$	−19.1372	−0.795316
$580$	0	0
$581$	6.60390	0.273976
$582$	0	0
$583$	21.8819	0.906256
$584$	0	0
$585$	−4.13992	−0.171165
$586$	0	0
$587$	28.2713	1.16688	0.583441	−	0.812155i	$-0.301706\pi$
$587$	28.2713	1.16688	0.583441	+	0.812155i	$0.301706\pi$
$588$	0	0
$589$	−41.8653	−1.72503
$590$	0	0
$591$	48.3576	1.98917
$592$	0	0
$593$	11.5579	0.474625	0.237312	−	0.971433i	$-0.423734\pi$
$593$	11.5579	0.474625	0.237312	+	0.971433i	$0.423734\pi$
$594$	0	0
$595$	4.44917	0.182398
$596$	0	0
$597$	−7.48341	−0.306275
$598$	0	0
$599$	7.51094	0.306889	0.153444	−	0.988157i	$-0.450963\pi$
$599$	7.51094	0.306889	0.153444	+	0.988157i	$0.450963\pi$
$600$	0	0
$601$	−21.0169	−0.857298	−0.428649	−	0.903471i	$-0.641010\pi$
$601$	−21.0169	−0.857298	−0.428649	+	0.903471i	$0.641010\pi$
$602$	0	0
$603$	−29.8808	−1.21684
$604$	0	0
$605$	3.00071	0.121996
$606$	0	0
$607$	−22.6722	−0.920237	−0.460119	−	0.887857i	$-0.652193\pi$
$607$	−22.6722	−0.920237	−0.460119	+	0.887857i	$0.652193\pi$
$608$	0	0
$609$	−12.6196	−0.511373
$610$	0	0
$611$	−0.623348	−0.0252180
$612$	0	0
$613$	−28.6091	−1.15551	−0.577755	−	0.816210i	$-0.696071\pi$
$613$	−28.6091	−1.15551	−0.577755	+	0.816210i	$0.696071\pi$
$614$	0	0
$615$	0.879440	0.0354624
$616$	0	0
$617$	−13.3771	−0.538541	−0.269271	−	0.963065i	$-0.586783\pi$
$617$	−13.3771	−0.538541	−0.269271	+	0.963065i	$0.586783\pi$
$618$	0	0
$619$	2.64262	0.106216	0.0531079	−	0.998589i	$-0.483087\pi$
$619$	2.64262	0.106216	0.0531079	+	0.998589i	$0.483087\pi$
$620$	0	0
$621$	7.87406	0.315975
$622$	0	0
$623$	−3.73139	−0.149495
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−36.1674	−1.44439
$628$	0	0
$629$	10.2179	0.407415
$630$	0	0
$631$	28.0974	1.11854	0.559270	−	0.828985i	$-0.311081\pi$
$631$	28.0974	1.11854	0.559270	+	0.828985i	$0.311081\pi$
$632$	0	0
$633$	1.70666	0.0678338
$634$	0	0
$635$	19.5514	0.775872
$636$	0	0
$637$	−12.4719	−0.494156
$638$	0	0
$639$	12.6549	0.500621
$640$	0	0
$641$	16.0345	0.633325	0.316662	−	0.948538i	$-0.397438\pi$
$641$	16.0345	0.633325	0.316662	+	0.948538i	$0.397438\pi$
$642$	0	0
$643$	−19.7772	−0.779936	−0.389968	−	0.920828i	$-0.627514\pi$
$643$	−19.7772	−0.779936	−0.389968	+	0.920828i	$0.627514\pi$
$644$	0	0
$645$	−12.9241	−0.508888
$646$	0	0
$647$	11.9084	0.468167	0.234083	−	0.972217i	$-0.424791\pi$
$647$	11.9084	0.468167	0.234083	+	0.972217i	$0.424791\pi$
$648$	0	0
$649$	22.7633	0.893538
$650$	0	0
$651$	14.2830	0.559793
$652$	0	0
$653$	−48.5089	−1.89830	−0.949149	−	0.314827i	$-0.898054\pi$
$653$	−48.5089	−1.89830	−0.949149	+	0.314827i	$0.898054\pi$
$654$	0	0
$655$	−3.98011	−0.155516
$656$	0	0
$657$	24.5942	0.959511
$658$	0	0
$659$	35.5188	1.38362	0.691809	−	0.722081i	$-0.256814\pi$
$659$	35.5188	1.38362	0.691809	+	0.722081i	$0.256814\pi$
$660$	0	0
$661$	25.7940	1.00327	0.501636	−	0.865079i	$-0.332732\pi$
$661$	25.7940	1.00327	0.501636	+	0.865079i	$0.332732\pi$
$662$	0	0
$663$	−23.2466	−0.902823
$664$	0	0
$665$	−4.87447	−0.189024
$666$	0	0
$667$	−24.7203	−0.957173
$668$	0	0
$669$	13.4232	0.518971
$670$	0	0
$671$	1.39142	0.0537153
$672$	0	0
$673$	24.7029	0.952226	0.476113	−	0.879384i	$-0.342045\pi$
$673$	24.7029	0.952226	0.476113	+	0.879384i	$0.342045\pi$
$674$	0	0
$675$	2.07640	0.0799205
$676$	0	0
$677$	−6.08457	−0.233849	−0.116924	−	0.993141i	$-0.537304\pi$
$677$	−6.08457	−0.233849	−0.116924	+	0.993141i	$0.537304\pi$
$678$	0	0
$679$	−4.88305	−0.187394
$680$	0	0
$681$	10.4957	0.402195
$682$	0	0
$683$	30.3875	1.16274	0.581372	−	0.813638i	$-0.302516\pi$
$683$	30.3875	1.16274	0.581372	+	0.813638i	$0.302516\pi$
$684$	0	0
$685$	−6.11881	−0.233788
$686$	0	0
$687$	24.8247	0.947123
$688$	0	0
$689$	−15.4090	−0.587038
$690$	0	0
$691$	−1.94639	−0.0740443	−0.0370221	−	0.999314i	$-0.511787\pi$
$691$	−1.94639	−0.0740443	−0.0370221	+	0.999314i	$0.511787\pi$
$692$	0	0
$693$	5.05022	0.191842
$694$	0	0
$695$	−13.4813	−0.511376
$696$	0	0
$697$	2.02117	0.0765573
$698$	0	0
$699$	6.37508	0.241128
$700$	0	0
$701$	−7.11248	−0.268635	−0.134317	−	0.990938i	$-0.542884\pi$
$701$	−7.11248	−0.268635	−0.134317	+	0.990938i	$0.542884\pi$
$702$	0	0
$703$	−11.1946	−0.422214
$704$	0	0
$705$	−0.705321	−0.0265639
$706$	0	0
$707$	−0.977362	−0.0367575
$708$	0	0
$709$	2.11804	0.0795448	0.0397724	−	0.999209i	$-0.487337\pi$
$709$	2.11804	0.0795448	0.0397724	+	0.999209i	$0.487337\pi$
$710$	0	0
$711$	9.26331	0.347401
$712$	0	0
$713$	27.9785	1.04780
$714$	0	0
$715$	5.63303	0.210664
$716$	0	0
$717$	60.2918	2.25164
$718$	0	0
$719$	48.2920	1.80099	0.900493	−	0.434870i	$-0.143206\pi$
$719$	48.2920	1.80099	0.900493	+	0.434870i	$0.143206\pi$
$720$	0	0
$721$	16.9418	0.630947
$722$	0	0
$723$	6.98456	0.259759
$724$	0	0
$725$	−6.51875	−0.242100
$726$	0	0
$727$	9.36707	0.347405	0.173703	−	0.984798i	$-0.444427\pi$
$727$	9.36707	0.347405	0.173703	+	0.984798i	$0.444427\pi$
$728$	0	0
$729$	−8.65061	−0.320393
$730$	0	0
$731$	−29.7029	−1.09860
$732$	0	0
$733$	−4.58496	−0.169349	−0.0846747	−	0.996409i	$-0.526985\pi$
$733$	−4.58496	−0.169349	−0.0846747	+	0.996409i	$0.526985\pi$
$734$	0	0
$735$	−14.1121	−0.520531
$736$	0	0
$737$	40.6577	1.49764
$738$	0	0
$739$	−5.66599	−0.208427	−0.104213	−	0.994555i	$-0.533233\pi$
$739$	−5.66599	−0.208427	−0.104213	+	0.994555i	$0.533233\pi$
$740$	0	0
$741$	25.4688	0.935618
$742$	0	0
$743$	−8.24957	−0.302647	−0.151324	−	0.988484i	$-0.548354\pi$
$743$	−8.24957	−0.302647	−0.151324	+	0.988484i	$0.548354\pi$
$744$	0	0
$745$	−9.91403	−0.363222
$746$	0	0
$747$	−15.9797	−0.584665
$748$	0	0
$749$	6.89764	0.252034
$750$	0	0
$751$	12.0354	0.439177	0.219588	−	0.975593i	$-0.429529\pi$
$751$	12.0354	0.439177	0.219588	+	0.975593i	$0.429529\pi$
$752$	0	0
$753$	−10.2620	−0.373967
$754$	0	0
$755$	−19.5149	−0.710220
$756$	0	0
$757$	7.02765	0.255424	0.127712	−	0.991811i	$-0.459237\pi$
$757$	7.02765	0.255424	0.127712	+	0.991811i	$0.459237\pi$
$758$	0	0
$759$	24.1706	0.877338
$760$	0	0
$761$	47.6983	1.72906	0.864531	−	0.502579i	$-0.167615\pi$
$761$	47.6983	1.72906	0.864531	+	0.502579i	$0.167615\pi$
$762$	0	0
$763$	−0.401812	−0.0145466
$764$	0	0
$765$	−10.7658	−0.389238
$766$	0	0
$767$	−16.0297	−0.578800
$768$	0	0
$769$	−8.87916	−0.320191	−0.160095	−	0.987102i	$-0.551180\pi$
$769$	−8.87916	−0.320191	−0.160095	+	0.987102i	$0.551180\pi$
$770$	0	0
$771$	−34.4642	−1.24120
$772$	0	0
$773$	24.2072	0.870672	0.435336	−	0.900268i	$-0.356630\pi$
$773$	24.2072	0.870672	0.435336	+	0.900268i	$0.356630\pi$
$774$	0	0
$775$	7.37796	0.265024
$776$	0	0
$777$	3.81922	0.137014
$778$	0	0
$779$	−2.21437	−0.0793382
$780$	0	0
$781$	−17.2191	−0.616146
$782$	0	0
$783$	−13.5355	−0.483720
$784$	0	0
$785$	14.4561	0.515960
$786$	0	0
$787$	−44.0646	−1.57073	−0.785366	−	0.619032i	$-0.787525\pi$
$787$	−44.0646	−1.57073	−0.785366	+	0.619032i	$0.787525\pi$
$788$	0	0
$789$	−12.3491	−0.439640
$790$	0	0
$791$	−15.5100	−0.551471
$792$	0	0
$793$	−0.979829	−0.0347947
$794$	0	0
$795$	−17.4354	−0.618370
$796$	0	0
$797$	41.5935	1.47332	0.736659	−	0.676264i	$-0.236402\pi$
$797$	41.5935	1.47332	0.736659	+	0.676264i	$0.236402\pi$
$798$	0	0
$799$	−1.62100	−0.0573469
$800$	0	0
$801$	9.02897	0.319023
$802$	0	0
$803$	−33.4644	−1.18093
$804$	0	0
$805$	3.25760	0.114815
$806$	0	0
$807$	5.66572	0.199443
$808$	0	0
$809$	−43.7414	−1.53787	−0.768933	−	0.639329i	$-0.779212\pi$
$809$	−43.7414	−1.53787	−0.768933	+	0.639329i	$0.779212\pi$
$810$	0	0
$811$	−28.3748	−0.996375	−0.498187	−	0.867069i	$-0.666001\pi$
$811$	−28.3748	−0.996375	−0.498187	+	0.867069i	$0.666001\pi$
$812$	0	0
$813$	−40.1413	−1.40782
$814$	0	0
$815$	−7.34353	−0.257233
$816$	0	0
$817$	32.5422	1.13851
$818$	0	0
$819$	−3.55632	−0.124268
$820$	0	0
$821$	43.1767	1.50688	0.753439	−	0.657518i	$-0.228393\pi$
$821$	43.1767	1.50688	0.753439	+	0.657518i	$0.228393\pi$
$822$	0	0
$823$	38.7898	1.35213	0.676064	−	0.736843i	$-0.263684\pi$
$823$	38.7898	1.35213	0.676064	+	0.736843i	$0.263684\pi$
$824$	0	0
$825$	6.37381	0.221907
$826$	0	0
$827$	37.8485	1.31612	0.658061	−	0.752965i	$-0.271377\pi$
$827$	37.8485	1.31612	0.658061	+	0.752965i	$0.271377\pi$
$828$	0	0
$829$	47.8473	1.66181	0.830903	−	0.556418i	$-0.187825\pi$
$829$	47.8473	1.66181	0.830903	+	0.556418i	$0.187825\pi$
$830$	0	0
$831$	5.15737	0.178907
$832$	0	0
$833$	−32.4330	−1.12374
$834$	0	0
$835$	20.3900	0.705626
$836$	0	0
$837$	15.3196	0.529521
$838$	0	0
$839$	−28.8039	−0.994421	−0.497211	−	0.867630i	$-0.665642\pi$
$839$	−28.8039	−0.994421	−0.497211	+	0.867630i	$0.665642\pi$
$840$	0	0
$841$	13.4942	0.465316
$842$	0	0
$843$	−56.8292	−1.95730
$844$	0	0
$845$	9.03327	0.310754
$846$	0	0
$847$	2.57770	0.0885708
$848$	0	0
$849$	−29.9665	−1.02845
$850$	0	0
$851$	7.48137	0.256458
$852$	0	0
$853$	−21.0098	−0.719361	−0.359681	−	0.933075i	$-0.617114\pi$
$853$	−21.0098	−0.719361	−0.359681	+	0.933075i	$0.617114\pi$
$854$	0	0
$855$	11.7949	0.403377
$856$	0	0
$857$	22.8051	0.779009	0.389504	−	0.921025i	$-0.372646\pi$
$857$	22.8051	0.779009	0.389504	+	0.921025i	$0.372646\pi$
$858$	0	0
$859$	−36.7265	−1.25309	−0.626546	−	0.779384i	$-0.715532\pi$
$859$	−36.7265	−1.25309	−0.626546	+	0.779384i	$0.715532\pi$
$860$	0	0
$861$	0.755466	0.0257462
$862$	0	0
$863$	−43.6110	−1.48454	−0.742268	−	0.670103i	$-0.766250\pi$
$863$	−43.6110	−1.48454	−0.742268	+	0.670103i	$0.766250\pi$
$864$	0	0
$865$	−7.97251	−0.271073
$866$	0	0
$867$	−22.1414	−0.751962
$868$	0	0
$869$	−12.6042	−0.427569
$870$	0	0
$871$	−28.6308	−0.970117
$872$	0	0
$873$	11.8157	0.399900
$874$	0	0
$875$	0.859031	0.0290406
$876$	0	0
$877$	14.6128	0.493440	0.246720	−	0.969087i	$-0.420647\pi$
$877$	14.6128	0.493440	0.246720	+	0.969087i	$0.420647\pi$
$878$	0	0
$879$	−61.1465	−2.06242
$880$	0	0
$881$	52.7052	1.77568	0.887842	−	0.460148i	$-0.152204\pi$
$881$	52.7052	1.77568	0.887842	+	0.460148i	$0.152204\pi$
$882$	0	0
$883$	−33.5988	−1.13069	−0.565345	−	0.824855i	$-0.691257\pi$
$883$	−33.5988	−1.13069	−0.565345	+	0.824855i	$0.691257\pi$
$884$	0	0
$885$	−18.1377	−0.609692
$886$	0	0
$887$	49.9796	1.67815	0.839076	−	0.544014i	$-0.183096\pi$
$887$	49.9796	1.67815	0.839076	+	0.544014i	$0.183096\pi$
$888$	0	0
$889$	16.7952	0.563294
$890$	0	0
$891$	30.8715	1.03423
$892$	0	0
$893$	1.77596	0.0594301
$894$	0	0
$895$	−8.83322	−0.295262
$896$	0	0
$897$	−17.0207	−0.568306
$898$	0	0
$899$	−48.0951	−1.60406
$900$	0	0
$901$	−40.0709	−1.33496
$902$	0	0
$903$	−11.1022	−0.369459
$904$	0	0
$905$	−13.4236	−0.446216
$906$	0	0
$907$	−29.1176	−0.966835	−0.483418	−	0.875390i	$-0.660605\pi$
$907$	−29.1176	−0.966835	−0.483418	+	0.875390i	$0.660605\pi$
$908$	0	0
$909$	2.36495	0.0784405
$910$	0	0
$911$	−3.84861	−0.127510	−0.0637551	−	0.997966i	$-0.520308\pi$
$911$	−3.84861	−0.127510	−0.0637551	+	0.997966i	$0.520308\pi$
$912$	0	0
$913$	21.7429	0.719585
$914$	0	0
$915$	−1.10868	−0.0366519
$916$	0	0
$917$	−3.41904	−0.112907
$918$	0	0
$919$	21.6254	0.713356	0.356678	−	0.934227i	$-0.383909\pi$
$919$	21.6254	0.713356	0.356678	+	0.934227i	$0.383909\pi$
$920$	0	0
$921$	7.81036	0.257360
$922$	0	0
$923$	12.1255	0.399116
$924$	0	0
$925$	1.97284	0.0648666
$926$	0	0
$927$	−40.9947	−1.34644
$928$	0	0
$929$	38.0092	1.24704	0.623520	−	0.781807i	$-0.285702\pi$
$929$	38.0092	1.24704	0.623520	+	0.781807i	$0.285702\pi$
$930$	0	0
$931$	35.5333	1.16456
$932$	0	0
$933$	−27.5711	−0.902637
$934$	0	0
$935$	14.6486	0.479060
$936$	0	0
$937$	−46.3898	−1.51549	−0.757744	−	0.652552i	$-0.773698\pi$
$937$	−46.3898	−1.51549	−0.757744	+	0.652552i	$0.773698\pi$
$938$	0	0
$939$	41.1667	1.34342
$940$	0	0
$941$	−0.871474	−0.0284092	−0.0142046	−	0.999899i	$-0.504522\pi$
$941$	−0.871474	−0.0284092	−0.0142046	+	0.999899i	$0.504522\pi$
$942$	0	0
$943$	1.47986	0.0481910
$944$	0	0
$945$	1.78369	0.0580234
$946$	0	0
$947$	6.14683	0.199745	0.0998725	−	0.995000i	$-0.468156\pi$
$947$	6.14683	0.199745	0.0998725	+	0.995000i	$0.468156\pi$
$948$	0	0
$949$	23.5653	0.764963
$950$	0	0
$951$	−53.6910	−1.74105
$952$	0	0
$953$	−39.5666	−1.28169	−0.640844	−	0.767671i	$-0.721416\pi$
$953$	−39.5666	−1.28169	−0.640844	+	0.767671i	$0.721416\pi$
$954$	0	0
$955$	8.57609	0.277516
$956$	0	0
$957$	−41.5493	−1.34310
$958$	0	0
$959$	−5.25625	−0.169733
$960$	0	0
$961$	23.4343	0.755945
$962$	0	0
$963$	−16.6904	−0.537842
$964$	0	0
$965$	−8.49192	−0.273365
$966$	0	0
$967$	44.3226	1.42532	0.712659	−	0.701510i	$-0.247491\pi$
$967$	44.3226	1.42532	0.712659	+	0.701510i	$0.247491\pi$
$968$	0	0
$969$	66.2310	2.12764
$970$	0	0
$971$	−45.5935	−1.46316	−0.731582	−	0.681753i	$-0.761218\pi$
$971$	−45.5935	−1.46316	−0.731582	+	0.681753i	$0.761218\pi$
$972$	0	0
$973$	−11.5809	−0.371266
$974$	0	0
$975$	−4.48838	−0.143743
$976$	0	0
$977$	38.4924	1.23148	0.615740	−	0.787950i	$-0.288857\pi$
$977$	38.4924	1.23148	0.615740	+	0.787950i	$0.288857\pi$
$978$	0	0
$979$	−12.2854	−0.392642
$980$	0	0
$981$	0.972276	0.0310424
$982$	0	0
$983$	28.2529	0.901127	0.450563	−	0.892744i	$-0.351223\pi$
$983$	28.2529	0.901127	0.450563	+	0.892744i	$0.351223\pi$
$984$	0	0
$985$	21.4581	0.683712
$986$	0	0
$987$	−0.605893	−0.0192858
$988$	0	0
$989$	−21.7479	−0.691543
$990$	0	0
$991$	−27.5720	−0.875854	−0.437927	−	0.899011i	$-0.644287\pi$
$991$	−27.5720	−0.875854	−0.437927	+	0.899011i	$0.644287\pi$
$992$	0	0
$993$	−33.9744	−1.07815
$994$	0	0
$995$	−3.32067	−0.105272
$996$	0	0
$997$	5.56767	0.176330	0.0881650	−	0.996106i	$-0.471900\pi$
$997$	5.56767	0.176330	0.0881650	+	0.996106i	$0.471900\pi$
$998$	0	0
$999$	4.09640	0.129604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.8	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.8	✓	35	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)

\(f(q)\)	\(=\)	\(q-2.25358 q^{3} -1.00000 q^{5} -0.859031 q^{7} +2.07862 q^{9} +O(q^{10})\)	\(q-2.25358 q^{3} -1.00000 q^{5} -0.859031 q^{7} +2.07862 q^{9} -2.82830 q^{11} +1.99167 q^{13} +2.25358 q^{15} +5.17928 q^{17} -5.67438 q^{19} +1.93590 q^{21} +3.79218 q^{23} +1.00000 q^{25} +2.07640 q^{27} -6.51875 q^{29} +7.37796 q^{31} +6.37381 q^{33} +0.859031 q^{35} +1.97284 q^{37} -4.48838 q^{39} +0.390241 q^{41} -5.73494 q^{43} -2.07862 q^{45} -0.312978 q^{47} -6.26207 q^{49} -11.6719 q^{51} -7.73676 q^{53} +2.82830 q^{55} +12.7877 q^{57} -8.04840 q^{59} -0.491964 q^{61} -1.78560 q^{63} -1.99167 q^{65} -14.3753 q^{67} -8.54598 q^{69} +6.08813 q^{71} +11.8320 q^{73} -2.25358 q^{75} +2.42960 q^{77} +4.45646 q^{79} -10.9152 q^{81} -7.68761 q^{83} -5.17928 q^{85} +14.6905 q^{87} +4.34372 q^{89} -1.71090 q^{91} -16.6268 q^{93} +5.67438 q^{95} +5.68437 q^{97} -5.87898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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