LMFDB - Embedded newform 8020.2.a.d.1.4

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:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.62437 q^{3} +1.00000 q^{5} -4.07774 q^{7} +3.88732 q^{9} +O(q^{10})$	$q-2.62437 q^{3} +1.00000 q^{5} -4.07774 q^{7} +3.88732 q^{9} +1.05635 q^{11} -4.00019 q^{13} -2.62437 q^{15} -5.19771 q^{17} -1.19846 q^{19} +10.7015 q^{21} +2.46052 q^{23} +1.00000 q^{25} -2.32865 q^{27} +3.14041 q^{29} +11.0654 q^{31} -2.77226 q^{33} -4.07774 q^{35} -0.856462 q^{37} +10.4980 q^{39} -8.16409 q^{41} -2.00070 q^{43} +3.88732 q^{45} +8.74288 q^{47} +9.62792 q^{49} +13.6407 q^{51} -7.04428 q^{53} +1.05635 q^{55} +3.14521 q^{57} -0.749784 q^{59} +13.9939 q^{61} -15.8514 q^{63} -4.00019 q^{65} +1.29961 q^{67} -6.45732 q^{69} +2.43501 q^{71} -1.60647 q^{73} -2.62437 q^{75} -4.30753 q^{77} +8.50903 q^{79} -5.55072 q^{81} +15.6217 q^{83} -5.19771 q^{85} -8.24159 q^{87} -6.38288 q^{89} +16.3117 q^{91} -29.0397 q^{93} -1.19846 q^{95} -2.90130 q^{97} +4.10638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * 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
$3$	−2.62437	−1.51518	−0.757590	−	0.652730i	$-0.773623\pi$
$3$	−2.62437	−1.51518	−0.757590	+	0.652730i	$0.773623\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.07774	−1.54124	−0.770620	−	0.637296i	$-0.780053\pi$
$7$	−4.07774	−1.54124	−0.770620	+	0.637296i	$0.780053\pi$
$8$	0	0
$9$	3.88732	1.29577
$10$	0	0
$11$	1.05635	0.318503	0.159251	−	0.987238i	$-0.449092\pi$
$11$	1.05635	0.318503	0.159251	+	0.987238i	$0.449092\pi$
$12$	0	0
$13$	−4.00019	−1.10945	−0.554727	−	0.832033i	$-0.687177\pi$
$13$	−4.00019	−1.10945	−0.554727	+	0.832033i	$0.687177\pi$
$14$	0	0
$15$	−2.62437	−0.677609
$16$	0	0
$17$	−5.19771	−1.26063	−0.630315	−	0.776339i	$-0.717074\pi$
$17$	−5.19771	−1.26063	−0.630315	+	0.776339i	$0.717074\pi$
$18$	0	0
$19$	−1.19846	−0.274946	−0.137473	−	0.990506i	$-0.543898\pi$
$19$	−1.19846	−0.274946	−0.137473	+	0.990506i	$0.543898\pi$
$20$	0	0
$21$	10.7015	2.33526
$22$	0	0
$23$	2.46052	0.513054	0.256527	−	0.966537i	$-0.417422\pi$
$23$	2.46052	0.513054	0.256527	+	0.966537i	$0.417422\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.32865	−0.448148
$28$	0	0
$29$	3.14041	0.583159	0.291580	−	0.956547i	$-0.405819\pi$
$29$	3.14041	0.583159	0.291580	+	0.956547i	$0.405819\pi$
$30$	0	0
$31$	11.0654	1.98740	0.993701	−	0.112064i	$-0.0357460\pi$
$31$	11.0654	1.98740	0.993701	+	0.112064i	$0.0357460\pi$
$32$	0	0
$33$	−2.77226	−0.482589
$34$	0	0
$35$	−4.07774	−0.689263
$36$	0	0
$37$	−0.856462	−0.140802	−0.0704008	−	0.997519i	$-0.522428\pi$
$37$	−0.856462	−0.140802	−0.0704008	+	0.997519i	$0.522428\pi$
$38$	0	0
$39$	10.4980	1.68102
$40$	0	0
$41$	−8.16409	−1.27502	−0.637508	−	0.770444i	$-0.720035\pi$
$41$	−8.16409	−1.27502	−0.637508	+	0.770444i	$0.720035\pi$
$42$	0	0
$43$	−2.00070	−0.305104	−0.152552	−	0.988295i	$-0.548749\pi$
$43$	−2.00070	−0.305104	−0.152552	+	0.988295i	$0.548749\pi$
$44$	0	0
$45$	3.88732	0.579487
$46$	0	0
$47$	8.74288	1.27528	0.637640	−	0.770335i	$-0.279911\pi$
$47$	8.74288	1.27528	0.637640	+	0.770335i	$0.279911\pi$
$48$	0	0
$49$	9.62792	1.37542
$50$	0	0
$51$	13.6407	1.91008
$52$	0	0
$53$	−7.04428	−0.967606	−0.483803	−	0.875177i	$-0.660745\pi$
$53$	−7.04428	−0.967606	−0.483803	+	0.875177i	$0.660745\pi$
$54$	0	0
$55$	1.05635	0.142439
$56$	0	0
$57$	3.14521	0.416593
$58$	0	0
$59$	−0.749784	−0.0976136	−0.0488068	−	0.998808i	$-0.515542\pi$
$59$	−0.749784	−0.0976136	−0.0488068	+	0.998808i	$0.515542\pi$
$60$	0	0
$61$	13.9939	1.79173	0.895866	−	0.444324i	$-0.146556\pi$
$61$	13.9939	1.79173	0.895866	+	0.444324i	$0.146556\pi$
$62$	0	0
$63$	−15.8514	−1.99709
$64$	0	0
$65$	−4.00019	−0.496163
$66$	0	0
$67$	1.29961	0.158772	0.0793861	−	0.996844i	$-0.474704\pi$
$67$	1.29961	0.158772	0.0793861	+	0.996844i	$0.474704\pi$
$68$	0	0
$69$	−6.45732	−0.777370
$70$	0	0
$71$	2.43501	0.288982	0.144491	−	0.989506i	$-0.453846\pi$
$71$	2.43501	0.288982	0.144491	+	0.989506i	$0.453846\pi$
$72$	0	0
$73$	−1.60647	−0.188023	−0.0940114	−	0.995571i	$-0.529969\pi$
$73$	−1.60647	−0.188023	−0.0940114	+	0.995571i	$0.529969\pi$
$74$	0	0
$75$	−2.62437	−0.303036
$76$	0	0
$77$	−4.30753	−0.490889
$78$	0	0
$79$	8.50903	0.957340	0.478670	−	0.877995i	$-0.341119\pi$
$79$	8.50903	0.957340	0.478670	+	0.877995i	$0.341119\pi$
$80$	0	0
$81$	−5.55072	−0.616747
$82$	0	0
$83$	15.6217	1.71471	0.857353	−	0.514729i	$-0.172107\pi$
$83$	15.6217	1.71471	0.857353	+	0.514729i	$0.172107\pi$
$84$	0	0
$85$	−5.19771	−0.563771
$86$	0	0
$87$	−8.24159	−0.883591
$88$	0	0
$89$	−6.38288	−0.676584	−0.338292	−	0.941041i	$-0.609849\pi$
$89$	−6.38288	−0.676584	−0.338292	+	0.941041i	$0.609849\pi$
$90$	0	0
$91$	16.3117	1.70993
$92$	0	0
$93$	−29.0397	−3.01127
$94$	0	0
$95$	−1.19846	−0.122960
$96$	0	0
$97$	−2.90130	−0.294583	−0.147291	−	0.989093i	$-0.547055\pi$
$97$	−2.90130	−0.294583	−0.147291	+	0.989093i	$0.547055\pi$
$98$	0	0
$99$	4.10638	0.412707
$100$	0	0
$101$	−0.365120	−0.0363308	−0.0181654	−	0.999835i	$-0.505783\pi$
$101$	−0.365120	−0.0363308	−0.0181654	+	0.999835i	$0.505783\pi$
$102$	0	0
$103$	2.83339	0.279183	0.139591	−	0.990209i	$-0.455421\pi$
$103$	2.83339	0.279183	0.139591	+	0.990209i	$0.455421\pi$
$104$	0	0
$105$	10.7015	1.04436
$106$	0	0
$107$	3.88836	0.375902	0.187951	−	0.982178i	$-0.439815\pi$
$107$	3.88836	0.375902	0.187951	+	0.982178i	$0.439815\pi$
$108$	0	0
$109$	0.316246	0.0302908	0.0151454	−	0.999885i	$-0.495179\pi$
$109$	0.316246	0.0302908	0.0151454	+	0.999885i	$0.495179\pi$
$110$	0	0
$111$	2.24767	0.213340
$112$	0	0
$113$	−18.3063	−1.72211	−0.861054	−	0.508513i	$-0.830195\pi$
$113$	−18.3063	−1.72211	−0.861054	+	0.508513i	$0.830195\pi$
$114$	0	0
$115$	2.46052	0.229445
$116$	0	0
$117$	−15.5500	−1.43760
$118$	0	0
$119$	21.1949	1.94293
$120$	0	0
$121$	−9.88412	−0.898556
$122$	0	0
$123$	21.4256	1.93188
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.0485	−1.51281	−0.756405	−	0.654104i	$-0.773046\pi$
$127$	−17.0485	−1.51281	−0.756405	+	0.654104i	$0.773046\pi$
$128$	0	0
$129$	5.25058	0.462287
$130$	0	0
$131$	−16.4851	−1.44031	−0.720157	−	0.693811i	$-0.755930\pi$
$131$	−16.4851	−1.44031	−0.720157	+	0.693811i	$0.755930\pi$
$132$	0	0
$133$	4.88701	0.423757
$134$	0	0
$135$	−2.32865	−0.200418
$136$	0	0
$137$	22.1816	1.89510	0.947552	−	0.319602i	$-0.103549\pi$
$137$	22.1816	1.89510	0.947552	+	0.319602i	$0.103549\pi$
$138$	0	0
$139$	−10.8376	−0.919233	−0.459617	−	0.888117i	$-0.652013\pi$
$139$	−10.8376	−0.919233	−0.459617	+	0.888117i	$0.652013\pi$
$140$	0	0
$141$	−22.9445	−1.93228
$142$	0	0
$143$	−4.22562	−0.353364
$144$	0	0
$145$	3.14041	0.260797
$146$	0	0
$147$	−25.2672	−2.08401
$148$	0	0
$149$	8.44366	0.691732	0.345866	−	0.938284i	$-0.387585\pi$
$149$	8.44366	0.691732	0.345866	+	0.938284i	$0.387585\pi$
$150$	0	0
$151$	7.26783	0.591448	0.295724	−	0.955273i	$-0.404439\pi$
$151$	7.26783	0.591448	0.295724	+	0.955273i	$0.404439\pi$
$152$	0	0
$153$	−20.2052	−1.63349
$154$	0	0
$155$	11.0654	0.888793
$156$	0	0
$157$	4.17278	0.333024	0.166512	−	0.986039i	$-0.446749\pi$
$157$	4.17278	0.333024	0.166512	+	0.986039i	$0.446749\pi$
$158$	0	0
$159$	18.4868	1.46610
$160$	0	0
$161$	−10.0334	−0.790739
$162$	0	0
$163$	7.92062	0.620391	0.310196	−	0.950673i	$-0.399605\pi$
$163$	7.92062	0.620391	0.310196	+	0.950673i	$0.399605\pi$
$164$	0	0
$165$	−2.77226	−0.215820
$166$	0	0
$167$	20.2969	1.57062	0.785311	−	0.619101i	$-0.212503\pi$
$167$	20.2969	1.57062	0.785311	+	0.619101i	$0.212503\pi$
$168$	0	0
$169$	3.00154	0.230887
$170$	0	0
$171$	−4.65880	−0.356267
$172$	0	0
$173$	16.1865	1.23064	0.615318	−	0.788279i	$-0.289027\pi$
$173$	16.1865	1.23064	0.615318	+	0.788279i	$0.289027\pi$
$174$	0	0
$175$	−4.07774	−0.308248
$176$	0	0
$177$	1.96771	0.147902
$178$	0	0
$179$	−3.87463	−0.289604	−0.144802	−	0.989461i	$-0.546254\pi$
$179$	−3.87463	−0.289604	−0.144802	+	0.989461i	$0.546254\pi$
$180$	0	0
$181$	17.6434	1.31143	0.655713	−	0.755011i	$-0.272368\pi$
$181$	17.6434	1.31143	0.655713	+	0.755011i	$0.272368\pi$
$182$	0	0
$183$	−36.7251	−2.71480
$184$	0	0
$185$	−0.856462	−0.0629684
$186$	0	0
$187$	−5.49062	−0.401514
$188$	0	0
$189$	9.49560	0.690703
$190$	0	0
$191$	24.0654	1.74131	0.870655	−	0.491894i	$-0.163695\pi$
$191$	24.0654	1.74131	0.870655	+	0.491894i	$0.163695\pi$
$192$	0	0
$193$	0.501524	0.0361005	0.0180502	−	0.999837i	$-0.494254\pi$
$193$	0.501524	0.0361005	0.0180502	+	0.999837i	$0.494254\pi$
$194$	0	0
$195$	10.4980	0.751776
$196$	0	0
$197$	−18.5551	−1.32200	−0.660998	−	0.750387i	$-0.729867\pi$
$197$	−18.5551	−1.32200	−0.660998	+	0.750387i	$0.729867\pi$
$198$	0	0
$199$	−9.99392	−0.708450	−0.354225	−	0.935160i	$-0.615255\pi$
$199$	−9.99392	−0.708450	−0.354225	+	0.935160i	$0.615255\pi$
$200$	0	0
$201$	−3.41065	−0.240569
$202$	0	0
$203$	−12.8058	−0.898788
$204$	0	0
$205$	−8.16409	−0.570204
$206$	0	0
$207$	9.56482	0.664801
$208$	0	0
$209$	−1.26600	−0.0875710
$210$	0	0
$211$	−20.1698	−1.38854	−0.694272	−	0.719713i	$-0.744274\pi$
$211$	−20.1698	−1.38854	−0.694272	+	0.719713i	$0.744274\pi$
$212$	0	0
$213$	−6.39036	−0.437860
$214$	0	0
$215$	−2.00070	−0.136447
$216$	0	0
$217$	−45.1217	−3.06306
$218$	0	0
$219$	4.21597	0.284889
$220$	0	0
$221$	20.7919	1.39861
$222$	0	0
$223$	7.57183	0.507047	0.253524	−	0.967329i	$-0.418410\pi$
$223$	7.57183	0.507047	0.253524	+	0.967329i	$0.418410\pi$
$224$	0	0
$225$	3.88732	0.259154
$226$	0	0
$227$	5.54688	0.368160	0.184080	−	0.982911i	$-0.441070\pi$
$227$	5.54688	0.368160	0.184080	+	0.982911i	$0.441070\pi$
$228$	0	0
$229$	11.5709	0.764627	0.382313	−	0.924033i	$-0.375127\pi$
$229$	11.5709	0.764627	0.382313	+	0.924033i	$0.375127\pi$
$230$	0	0
$231$	11.3046	0.743785
$232$	0	0
$233$	−16.2306	−1.06330	−0.531649	−	0.846965i	$-0.678428\pi$
$233$	−16.2306	−1.06330	−0.531649	+	0.846965i	$0.678428\pi$
$234$	0	0
$235$	8.74288	0.570322
$236$	0	0
$237$	−22.3308	−1.45054
$238$	0	0
$239$	−12.4871	−0.807723	−0.403861	−	0.914820i	$-0.632332\pi$
$239$	−12.4871	−0.807723	−0.403861	+	0.914820i	$0.632332\pi$
$240$	0	0
$241$	−26.0067	−1.67524	−0.837620	−	0.546253i	$-0.816054\pi$
$241$	−26.0067	−1.67524	−0.837620	+	0.546253i	$0.816054\pi$
$242$	0	0
$243$	21.5531	1.38263
$244$	0	0
$245$	9.62792	0.615105
$246$	0	0
$247$	4.79407	0.305040
$248$	0	0
$249$	−40.9972	−2.59809
$250$	0	0
$251$	−7.94957	−0.501773	−0.250886	−	0.968017i	$-0.580722\pi$
$251$	−7.94957	−0.501773	−0.250886	+	0.968017i	$0.580722\pi$
$252$	0	0
$253$	2.59918	0.163409
$254$	0	0
$255$	13.6407	0.854215
$256$	0	0
$257$	24.9177	1.55432	0.777160	−	0.629303i	$-0.216659\pi$
$257$	24.9177	1.55432	0.777160	+	0.629303i	$0.216659\pi$
$258$	0	0
$259$	3.49243	0.217009
$260$	0	0
$261$	12.2078	0.755641
$262$	0	0
$263$	3.95851	0.244092	0.122046	−	0.992524i	$-0.461054\pi$
$263$	3.95851	0.244092	0.122046	+	0.992524i	$0.461054\pi$
$264$	0	0
$265$	−7.04428	−0.432727
$266$	0	0
$267$	16.7510	1.02515
$268$	0	0
$269$	−11.8277	−0.721150	−0.360575	−	0.932730i	$-0.617420\pi$
$269$	−11.8277	−0.721150	−0.360575	+	0.932730i	$0.617420\pi$
$270$	0	0
$271$	−3.19991	−0.194381	−0.0971904	−	0.995266i	$-0.530986\pi$
$271$	−3.19991	−0.194381	−0.0971904	+	0.995266i	$0.530986\pi$
$272$	0	0
$273$	−42.8080	−2.59086
$274$	0	0
$275$	1.05635	0.0637005
$276$	0	0
$277$	−18.0357	−1.08366	−0.541831	−	0.840488i	$-0.682269\pi$
$277$	−18.0357	−1.08366	−0.541831	+	0.840488i	$0.682269\pi$
$278$	0	0
$279$	43.0147	2.57522
$280$	0	0
$281$	25.4996	1.52118	0.760588	−	0.649235i	$-0.224911\pi$
$281$	25.4996	1.52118	0.760588	+	0.649235i	$0.224911\pi$
$282$	0	0
$283$	−25.3551	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$283$	−25.3551	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$284$	0	0
$285$	3.14521	0.186306
$286$	0	0
$287$	33.2910	1.96510
$288$	0	0
$289$	10.0162	0.589190
$290$	0	0
$291$	7.61409	0.446346
$292$	0	0
$293$	−8.36079	−0.488443	−0.244221	−	0.969720i	$-0.578532\pi$
$293$	−8.36079	−0.488443	−0.244221	+	0.969720i	$0.578532\pi$
$294$	0	0
$295$	−0.749784	−0.0436541
$296$	0	0
$297$	−2.45987	−0.142736
$298$	0	0
$299$	−9.84256	−0.569210
$300$	0	0
$301$	8.15832	0.470238
$302$	0	0
$303$	0.958209	0.0550477
$304$	0	0
$305$	13.9939	0.801287
$306$	0	0
$307$	−21.6431	−1.23524	−0.617618	−	0.786478i	$-0.711902\pi$
$307$	−21.6431	−1.23524	−0.617618	+	0.786478i	$0.711902\pi$
$308$	0	0
$309$	−7.43587	−0.423012
$310$	0	0
$311$	32.6714	1.85263	0.926314	−	0.376753i	$-0.122959\pi$
$311$	32.6714	1.85263	0.926314	+	0.376753i	$0.122959\pi$
$312$	0	0
$313$	21.9634	1.24145	0.620723	−	0.784030i	$-0.286839\pi$
$313$	21.9634	1.24145	0.620723	+	0.784030i	$0.286839\pi$
$314$	0	0
$315$	−15.8514	−0.893128
$316$	0	0
$317$	−5.23398	−0.293970	−0.146985	−	0.989139i	$-0.546957\pi$
$317$	−5.23398	−0.293970	−0.146985	+	0.989139i	$0.546957\pi$
$318$	0	0
$319$	3.31738	0.185738
$320$	0	0
$321$	−10.2045	−0.569560
$322$	0	0
$323$	6.22926	0.346605
$324$	0	0
$325$	−4.00019	−0.221891
$326$	0	0
$327$	−0.829945	−0.0458961
$328$	0	0
$329$	−35.6511	−1.96551
$330$	0	0
$331$	−1.88977	−0.103871	−0.0519357	−	0.998650i	$-0.516539\pi$
$331$	−1.88977	−0.103871	−0.0519357	+	0.998650i	$0.516539\pi$
$332$	0	0
$333$	−3.32934	−0.182447
$334$	0	0
$335$	1.29961	0.0710051
$336$	0	0
$337$	−2.93863	−0.160077	−0.0800387	−	0.996792i	$-0.525504\pi$
$337$	−2.93863	−0.160077	−0.0800387	+	0.996792i	$0.525504\pi$
$338$	0	0
$339$	48.0424	2.60931
$340$	0	0
$341$	11.6890	0.632993
$342$	0	0
$343$	−10.7160	−0.578608
$344$	0	0
$345$	−6.45732	−0.347650
$346$	0	0
$347$	−13.3661	−0.717532	−0.358766	−	0.933427i	$-0.616802\pi$
$347$	−13.3661	−0.717532	−0.358766	+	0.933427i	$0.616802\pi$
$348$	0	0
$349$	11.5005	0.615610	0.307805	−	0.951449i	$-0.400406\pi$
$349$	11.5005	0.615610	0.307805	+	0.951449i	$0.400406\pi$
$350$	0	0
$351$	9.31503	0.497200
$352$	0	0
$353$	−8.64343	−0.460043	−0.230022	−	0.973185i	$-0.573880\pi$
$353$	−8.64343	−0.460043	−0.230022	+	0.973185i	$0.573880\pi$
$354$	0	0
$355$	2.43501	0.129237
$356$	0	0
$357$	−55.6232	−2.94389
$358$	0	0
$359$	−35.5991	−1.87885	−0.939424	−	0.342759i	$-0.888639\pi$
$359$	−35.5991	−1.87885	−0.939424	+	0.342759i	$0.888639\pi$
$360$	0	0
$361$	−17.5637	−0.924405
$362$	0	0
$363$	25.9396	1.36147
$364$	0	0
$365$	−1.60647	−0.0840864
$366$	0	0
$367$	6.96952	0.363806	0.181903	−	0.983316i	$-0.441774\pi$
$367$	6.96952	0.363806	0.181903	+	0.983316i	$0.441774\pi$
$368$	0	0
$369$	−31.7364	−1.65213
$370$	0	0
$371$	28.7247	1.49131
$372$	0	0
$373$	5.30785	0.274830	0.137415	−	0.990514i	$-0.456121\pi$
$373$	5.30785	0.274830	0.137415	+	0.990514i	$0.456121\pi$
$374$	0	0
$375$	−2.62437	−0.135522
$376$	0	0
$377$	−12.5622	−0.646988
$378$	0	0
$379$	−2.62851	−0.135017	−0.0675087	−	0.997719i	$-0.521505\pi$
$379$	−2.62851	−0.135017	−0.0675087	+	0.997719i	$0.521505\pi$
$380$	0	0
$381$	44.7415	2.29218
$382$	0	0
$383$	7.37772	0.376984	0.188492	−	0.982075i	$-0.439640\pi$
$383$	7.37772	0.376984	0.188492	+	0.982075i	$0.439640\pi$
$384$	0	0
$385$	−4.30753	−0.219532
$386$	0	0
$387$	−7.77735	−0.395345
$388$	0	0
$389$	−34.0789	−1.72787	−0.863935	−	0.503604i	$-0.832007\pi$
$389$	−34.0789	−1.72787	−0.863935	+	0.503604i	$0.832007\pi$
$390$	0	0
$391$	−12.7891	−0.646772
$392$	0	0
$393$	43.2631	2.18234
$394$	0	0
$395$	8.50903	0.428136
$396$	0	0
$397$	3.54313	0.177825	0.0889123	−	0.996039i	$-0.471661\pi$
$397$	3.54313	0.177825	0.0889123	+	0.996039i	$0.471661\pi$
$398$	0	0
$399$	−12.8253	−0.642069
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−44.2637	−2.20493
$404$	0	0
$405$	−5.55072	−0.275818
$406$	0	0
$407$	−0.904727	−0.0448456
$408$	0	0
$409$	23.0466	1.13958	0.569790	−	0.821790i	$-0.307024\pi$
$409$	23.0466	1.13958	0.569790	+	0.821790i	$0.307024\pi$
$410$	0	0
$411$	−58.2128	−2.87142
$412$	0	0
$413$	3.05742	0.150446
$414$	0	0
$415$	15.6217	0.766840
$416$	0	0
$417$	28.4419	1.39280
$418$	0	0
$419$	−2.67552	−0.130708	−0.0653539	−	0.997862i	$-0.520818\pi$
$419$	−2.67552	−0.130708	−0.0653539	+	0.997862i	$0.520818\pi$
$420$	0	0
$421$	0.926580	0.0451587	0.0225794	−	0.999745i	$-0.492812\pi$
$421$	0.926580	0.0451587	0.0225794	+	0.999745i	$0.492812\pi$
$422$	0	0
$423$	33.9863	1.65247
$424$	0	0
$425$	−5.19771	−0.252126
$426$	0	0
$427$	−57.0633	−2.76149
$428$	0	0
$429$	11.0896	0.535410
$430$	0	0
$431$	−26.2892	−1.26631	−0.633153	−	0.774027i	$-0.718239\pi$
$431$	−26.2892	−1.26631	−0.633153	+	0.774027i	$0.718239\pi$
$432$	0	0
$433$	−38.2560	−1.83846	−0.919232	−	0.393716i	$-0.871189\pi$
$433$	−38.2560	−1.83846	−0.919232	+	0.393716i	$0.871189\pi$
$434$	0	0
$435$	−8.24159	−0.395154
$436$	0	0
$437$	−2.94884	−0.141062
$438$	0	0
$439$	−16.2952	−0.777730	−0.388865	−	0.921295i	$-0.627133\pi$
$439$	−16.2952	−0.777730	−0.388865	+	0.921295i	$0.627133\pi$
$440$	0	0
$441$	37.4268	1.78223
$442$	0	0
$443$	41.0777	1.95166	0.975830	−	0.218532i	$-0.0701269\pi$
$443$	41.0777	1.95166	0.975830	+	0.218532i	$0.0701269\pi$
$444$	0	0
$445$	−6.38288	−0.302578
$446$	0	0
$447$	−22.1593	−1.04810
$448$	0	0
$449$	−10.8776	−0.513348	−0.256674	−	0.966498i	$-0.582627\pi$
$449$	−10.8776	−0.513348	−0.256674	+	0.966498i	$0.582627\pi$
$450$	0	0
$451$	−8.62416	−0.406096
$452$	0	0
$453$	−19.0735	−0.896150
$454$	0	0
$455$	16.3117	0.764705
$456$	0	0
$457$	−40.6931	−1.90354	−0.951771	−	0.306810i	$-0.900739\pi$
$457$	−40.6931	−1.90354	−0.951771	+	0.306810i	$0.900739\pi$
$458$	0	0
$459$	12.1036	0.564949
$460$	0	0
$461$	−16.7337	−0.779366	−0.389683	−	0.920949i	$-0.627415\pi$
$461$	−16.7337	−0.779366	−0.389683	+	0.920949i	$0.627415\pi$
$462$	0	0
$463$	−11.4077	−0.530163	−0.265081	−	0.964226i	$-0.585399\pi$
$463$	−11.4077	−0.530163	−0.265081	+	0.964226i	$0.585399\pi$
$464$	0	0
$465$	−29.0397	−1.34668
$466$	0	0
$467$	1.45950	0.0675374	0.0337687	−	0.999430i	$-0.489249\pi$
$467$	1.45950	0.0675374	0.0337687	+	0.999430i	$0.489249\pi$
$468$	0	0
$469$	−5.29945	−0.244706
$470$	0	0
$471$	−10.9509	−0.504592
$472$	0	0
$473$	−2.11345	−0.0971764
$474$	0	0
$475$	−1.19846	−0.0549892
$476$	0	0
$477$	−27.3833	−1.25380
$478$	0	0
$479$	−35.2640	−1.61125	−0.805626	−	0.592424i	$-0.798171\pi$
$479$	−35.2640	−1.61125	−0.805626	+	0.592424i	$0.798171\pi$
$480$	0	0
$481$	3.42601	0.156213
$482$	0	0
$483$	26.3312	1.19811
$484$	0	0
$485$	−2.90130	−0.131741
$486$	0	0
$487$	−29.7000	−1.34584	−0.672918	−	0.739717i	$-0.734959\pi$
$487$	−29.7000	−1.34584	−0.672918	+	0.739717i	$0.734959\pi$
$488$	0	0
$489$	−20.7866	−0.940005
$490$	0	0
$491$	−13.1321	−0.592644	−0.296322	−	0.955088i	$-0.595760\pi$
$491$	−13.1321	−0.592644	−0.296322	+	0.955088i	$0.595760\pi$
$492$	0	0
$493$	−16.3229	−0.735148
$494$	0	0
$495$	4.10638	0.184568
$496$	0	0
$497$	−9.92931	−0.445391
$498$	0	0
$499$	−26.4620	−1.18460	−0.592302	−	0.805716i	$-0.701781\pi$
$499$	−26.4620	−1.18460	−0.592302	+	0.805716i	$0.701781\pi$
$500$	0	0
$501$	−53.2666	−2.37978
$502$	0	0
$503$	−16.5219	−0.736676	−0.368338	−	0.929692i	$-0.620073\pi$
$503$	−16.5219	−0.736676	−0.368338	+	0.929692i	$0.620073\pi$
$504$	0	0
$505$	−0.365120	−0.0162476
$506$	0	0
$507$	−7.87714	−0.349836
$508$	0	0
$509$	25.4944	1.13002	0.565010	−	0.825084i	$-0.308872\pi$
$509$	25.4944	1.13002	0.565010	+	0.825084i	$0.308872\pi$
$510$	0	0
$511$	6.55075	0.289788
$512$	0	0
$513$	2.79079	0.123216
$514$	0	0
$515$	2.83339	0.124854
$516$	0	0
$517$	9.23557	0.406180
$518$	0	0
$519$	−42.4794	−1.86464
$520$	0	0
$521$	−6.19890	−0.271579	−0.135789	−	0.990738i	$-0.543357\pi$
$521$	−6.19890	−0.271579	−0.135789	+	0.990738i	$0.543357\pi$
$522$	0	0
$523$	6.95078	0.303936	0.151968	−	0.988385i	$-0.451439\pi$
$523$	6.95078	0.303936	0.151968	+	0.988385i	$0.451439\pi$
$524$	0	0
$525$	10.7015	0.467051
$526$	0	0
$527$	−57.5147	−2.50538
$528$	0	0
$529$	−16.9458	−0.736775
$530$	0	0
$531$	−2.91465	−0.126485
$532$	0	0
$533$	32.6579	1.41457
$534$	0	0
$535$	3.88836	0.168109
$536$	0	0
$537$	10.1685	0.438802
$538$	0	0
$539$	10.1705	0.438074
$540$	0	0
$541$	10.4806	0.450597	0.225299	−	0.974290i	$-0.427664\pi$
$541$	10.4806	0.450597	0.225299	+	0.974290i	$0.427664\pi$
$542$	0	0
$543$	−46.3029	−1.98705
$544$	0	0
$545$	0.316246	0.0135465
$546$	0	0
$547$	37.5729	1.60650	0.803250	−	0.595642i	$-0.203102\pi$
$547$	37.5729	1.60650	0.803250	+	0.595642i	$0.203102\pi$
$548$	0	0
$549$	54.3986	2.32168
$550$	0	0
$551$	−3.76366	−0.160337
$552$	0	0
$553$	−34.6976	−1.47549
$554$	0	0
$555$	2.24767	0.0954084
$556$	0	0
$557$	−17.1404	−0.726264	−0.363132	−	0.931738i	$-0.618293\pi$
$557$	−17.1404	−0.726264	−0.363132	+	0.931738i	$0.618293\pi$
$558$	0	0
$559$	8.00318	0.338499
$560$	0	0
$561$	14.4094	0.608366
$562$	0	0
$563$	20.3748	0.858693	0.429347	−	0.903140i	$-0.358744\pi$
$563$	20.3748	0.858693	0.429347	+	0.903140i	$0.358744\pi$
$564$	0	0
$565$	−18.3063	−0.770150
$566$	0	0
$567$	22.6344	0.950554
$568$	0	0
$569$	10.2870	0.431254	0.215627	−	0.976476i	$-0.430820\pi$
$569$	10.2870	0.431254	0.215627	+	0.976476i	$0.430820\pi$
$570$	0	0
$571$	27.5624	1.15345	0.576725	−	0.816939i	$-0.304331\pi$
$571$	27.5624	1.15345	0.576725	+	0.816939i	$0.304331\pi$
$572$	0	0
$573$	−63.1565	−2.63840
$574$	0	0
$575$	2.46052	0.102611
$576$	0	0
$577$	9.17130	0.381806	0.190903	−	0.981609i	$-0.438858\pi$
$577$	9.17130	0.381806	0.190903	+	0.981609i	$0.438858\pi$
$578$	0	0
$579$	−1.31618	−0.0546988
$580$	0	0
$581$	−63.7012	−2.64277
$582$	0	0
$583$	−7.44125	−0.308185
$584$	0	0
$585$	−15.5500	−0.642914
$586$	0	0
$587$	7.33623	0.302799	0.151399	−	0.988473i	$-0.451622\pi$
$587$	7.33623	0.302799	0.151399	+	0.988473i	$0.451622\pi$
$588$	0	0
$589$	−13.2614	−0.546428
$590$	0	0
$591$	48.6955	2.00306
$592$	0	0
$593$	−38.4291	−1.57810	−0.789048	−	0.614332i	$-0.789426\pi$
$593$	−38.4291	−1.57810	−0.789048	+	0.614332i	$0.789426\pi$
$594$	0	0
$595$	21.1949	0.868906
$596$	0	0
$597$	26.2278	1.07343
$598$	0	0
$599$	−22.1296	−0.904191	−0.452096	−	0.891970i	$-0.649323\pi$
$599$	−22.1296	−0.904191	−0.452096	+	0.891970i	$0.649323\pi$
$600$	0	0
$601$	15.4945	0.632032	0.316016	−	0.948754i	$-0.397655\pi$
$601$	15.4945	0.632032	0.316016	+	0.948754i	$0.397655\pi$
$602$	0	0
$603$	5.05198	0.205733
$604$	0	0
$605$	−9.88412	−0.401847
$606$	0	0
$607$	11.6227	0.471752	0.235876	−	0.971783i	$-0.424204\pi$
$607$	11.6227	0.471752	0.235876	+	0.971783i	$0.424204\pi$
$608$	0	0
$609$	33.6070	1.36183
$610$	0	0
$611$	−34.9732	−1.41486
$612$	0	0
$613$	24.3950	0.985307	0.492653	−	0.870226i	$-0.336027\pi$
$613$	24.3950	0.985307	0.492653	+	0.870226i	$0.336027\pi$
$614$	0	0
$615$	21.4256	0.863963
$616$	0	0
$617$	6.11556	0.246203	0.123102	−	0.992394i	$-0.460716\pi$
$617$	6.11556	0.246203	0.123102	+	0.992394i	$0.460716\pi$
$618$	0	0
$619$	21.0315	0.845329	0.422664	−	0.906286i	$-0.361095\pi$
$619$	21.0315	0.845329	0.422664	+	0.906286i	$0.361095\pi$
$620$	0	0
$621$	−5.72968	−0.229924
$622$	0	0
$623$	26.0277	1.04278
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.32245	0.132686
$628$	0	0
$629$	4.45165	0.177499
$630$	0	0
$631$	8.27036	0.329238	0.164619	−	0.986357i	$-0.447361\pi$
$631$	8.27036	0.329238	0.164619	+	0.986357i	$0.447361\pi$
$632$	0	0
$633$	52.9329	2.10390
$634$	0	0
$635$	−17.0485	−0.676549
$636$	0	0
$637$	−38.5135	−1.52596
$638$	0	0
$639$	9.46564	0.374455
$640$	0	0
$641$	40.5814	1.60287	0.801434	−	0.598083i	$-0.204071\pi$
$641$	40.5814	1.60287	0.801434	+	0.598083i	$0.204071\pi$
$642$	0	0
$643$	−47.7642	−1.88364	−0.941819	−	0.336121i	$-0.890885\pi$
$643$	−47.7642	−1.88364	−0.941819	+	0.336121i	$0.890885\pi$
$644$	0	0
$645$	5.25058	0.206741
$646$	0	0
$647$	−21.1129	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$647$	−21.1129	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$648$	0	0
$649$	−0.792037	−0.0310902
$650$	0	0
$651$	118.416	4.64109
$652$	0	0
$653$	−29.3830	−1.14984	−0.574922	−	0.818208i	$-0.694968\pi$
$653$	−29.3830	−1.14984	−0.574922	+	0.818208i	$0.694968\pi$
$654$	0	0
$655$	−16.4851	−0.644128
$656$	0	0
$657$	−6.24485	−0.243635
$658$	0	0
$659$	−9.16198	−0.356900	−0.178450	−	0.983949i	$-0.557108\pi$
$659$	−9.16198	−0.356900	−0.178450	+	0.983949i	$0.557108\pi$
$660$	0	0
$661$	−20.1781	−0.784838	−0.392419	−	0.919787i	$-0.628362\pi$
$661$	−20.1781	−0.784838	−0.392419	+	0.919787i	$0.628362\pi$
$662$	0	0
$663$	−54.5655	−2.11915
$664$	0	0
$665$	4.88701	0.189510
$666$	0	0
$667$	7.72704	0.299192
$668$	0	0
$669$	−19.8713	−0.768268
$670$	0	0
$671$	14.7825	0.570671
$672$	0	0
$673$	−11.8557	−0.457003	−0.228502	−	0.973544i	$-0.573383\pi$
$673$	−11.8557	−0.457003	−0.228502	+	0.973544i	$0.573383\pi$
$674$	0	0
$675$	−2.32865	−0.0896296
$676$	0	0
$677$	−32.9717	−1.26721	−0.633603	−	0.773658i	$-0.718425\pi$
$677$	−32.9717	−1.26721	−0.633603	+	0.773658i	$0.718425\pi$
$678$	0	0
$679$	11.8307	0.454022
$680$	0	0
$681$	−14.5571	−0.557828
$682$	0	0
$683$	−0.0150951	−0.000577599 0	−0.000288799	−	1.00000i	$-0.500092\pi$
$683$	−0.0150951	−0.000577599 0	−0.000288799		1.00000i	$0.500092\pi$
$684$	0	0
$685$	22.1816	0.847516
$686$	0	0
$687$	−30.3663	−1.15855
$688$	0	0
$689$	28.1785	1.07351
$690$	0	0
$691$	9.22951	0.351107	0.175554	−	0.984470i	$-0.443828\pi$
$691$	9.22951	0.351107	0.175554	+	0.984470i	$0.443828\pi$
$692$	0	0
$693$	−16.7447	−0.636080
$694$	0	0
$695$	−10.8376	−0.411093
$696$	0	0
$697$	42.4346	1.60732
$698$	0	0
$699$	42.5950	1.61109
$700$	0	0
$701$	10.3421	0.390617	0.195308	−	0.980742i	$-0.437429\pi$
$701$	10.3421	0.390617	0.195308	+	0.980742i	$0.437429\pi$
$702$	0	0
$703$	1.02644	0.0387128
$704$	0	0
$705$	−22.9445	−0.864141
$706$	0	0
$707$	1.48886	0.0559944
$708$	0	0
$709$	−52.4354	−1.96925	−0.984627	−	0.174670i	$-0.944114\pi$
$709$	−52.4354	−1.96925	−0.984627	+	0.174670i	$0.944114\pi$
$710$	0	0
$711$	33.0773	1.24049
$712$	0	0
$713$	27.2266	1.01964
$714$	0	0
$715$	−4.22562	−0.158029
$716$	0	0
$717$	32.7707	1.22385
$718$	0	0
$719$	−18.4086	−0.686524	−0.343262	−	0.939240i	$-0.611532\pi$
$719$	−18.4086	−0.686524	−0.343262	+	0.939240i	$0.611532\pi$
$720$	0	0
$721$	−11.5538	−0.430287
$722$	0	0
$723$	68.2513	2.53829
$724$	0	0
$725$	3.14041	0.116632
$726$	0	0
$727$	29.0503	1.07742	0.538708	−	0.842493i	$-0.318913\pi$
$727$	29.0503	1.07742	0.538708	+	0.842493i	$0.318913\pi$
$728$	0	0
$729$	−39.9111	−1.47819
$730$	0	0
$731$	10.3991	0.384623
$732$	0	0
$733$	15.5985	0.576142	0.288071	−	0.957609i	$-0.406986\pi$
$733$	15.5985	0.576142	0.288071	+	0.957609i	$0.406986\pi$
$734$	0	0
$735$	−25.2672	−0.931996
$736$	0	0
$737$	1.37284	0.0505693
$738$	0	0
$739$	−46.0476	−1.69389	−0.846944	−	0.531682i	$-0.821560\pi$
$739$	−46.0476	−1.69389	−0.846944	+	0.531682i	$0.821560\pi$
$740$	0	0
$741$	−12.5814	−0.462190
$742$	0	0
$743$	−7.70978	−0.282844	−0.141422	−	0.989949i	$-0.545167\pi$
$743$	−7.70978	−0.282844	−0.141422	+	0.989949i	$0.545167\pi$
$744$	0	0
$745$	8.44366	0.309352
$746$	0	0
$747$	60.7266	2.22187
$748$	0	0
$749$	−15.8557	−0.579355
$750$	0	0
$751$	−5.75074	−0.209847	−0.104924	−	0.994480i	$-0.533460\pi$
$751$	−5.75074	−0.209847	−0.104924	+	0.994480i	$0.533460\pi$
$752$	0	0
$753$	20.8626	0.760276
$754$	0	0
$755$	7.26783	0.264504
$756$	0	0
$757$	−5.45621	−0.198309	−0.0991546	−	0.995072i	$-0.531614\pi$
$757$	−5.45621	−0.198309	−0.0991546	+	0.995072i	$0.531614\pi$
$758$	0	0
$759$	−6.82121	−0.247594
$760$	0	0
$761$	25.9380	0.940253	0.470126	−	0.882599i	$-0.344208\pi$
$761$	25.9380	0.940253	0.470126	+	0.882599i	$0.344208\pi$
$762$	0	0
$763$	−1.28957	−0.0466854
$764$	0	0
$765$	−20.2052	−0.730519
$766$	0	0
$767$	2.99928	0.108298
$768$	0	0
$769$	−39.5689	−1.42689	−0.713445	−	0.700711i	$-0.752866\pi$
$769$	−39.5689	−1.42689	−0.713445	+	0.700711i	$0.752866\pi$
$770$	0	0
$771$	−65.3931	−2.35508
$772$	0	0
$773$	−42.3933	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$773$	−42.3933	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$774$	0	0
$775$	11.0654	0.397480
$776$	0	0
$777$	−9.16542	−0.328808
$778$	0	0
$779$	9.78434	0.350560
$780$	0	0
$781$	2.57223	0.0920415
$782$	0	0
$783$	−7.31290	−0.261342
$784$	0	0
$785$	4.17278	0.148933
$786$	0	0
$787$	28.6406	1.02093	0.510464	−	0.859899i	$-0.329474\pi$
$787$	28.6406	1.02093	0.510464	+	0.859899i	$0.329474\pi$
$788$	0	0
$789$	−10.3886	−0.369844
$790$	0	0
$791$	74.6481	2.65418
$792$	0	0
$793$	−55.9782	−1.98784
$794$	0	0
$795$	18.4868	0.655659
$796$	0	0
$797$	34.2463	1.21307	0.606534	−	0.795058i	$-0.292560\pi$
$797$	34.2463	1.21307	0.606534	+	0.795058i	$0.292560\pi$
$798$	0	0
$799$	−45.4430	−1.60766
$800$	0	0
$801$	−24.8123	−0.876699
$802$	0	0
$803$	−1.69700	−0.0598857
$804$	0	0
$805$	−10.0334	−0.353629
$806$	0	0
$807$	31.0404	1.09267
$808$	0	0
$809$	10.1258	0.356003	0.178002	−	0.984030i	$-0.443037\pi$
$809$	10.1258	0.356003	0.178002	+	0.984030i	$0.443037\pi$
$810$	0	0
$811$	11.3131	0.397255	0.198628	−	0.980075i	$-0.436352\pi$
$811$	11.3131	0.397255	0.198628	+	0.980075i	$0.436352\pi$
$812$	0	0
$813$	8.39775	0.294522
$814$	0	0
$815$	7.92062	0.277447
$816$	0	0
$817$	2.39776	0.0838870
$818$	0	0
$819$	63.4088	2.21568
$820$	0	0
$821$	−35.9726	−1.25545	−0.627726	−	0.778434i	$-0.716014\pi$
$821$	−35.9726	−1.25545	−0.627726	+	0.778434i	$0.716014\pi$
$822$	0	0
$823$	53.9251	1.87971	0.939855	−	0.341572i	$-0.110959\pi$
$823$	53.9251	1.87971	0.939855	+	0.341572i	$0.110959\pi$
$824$	0	0
$825$	−2.77226	−0.0965178
$826$	0	0
$827$	10.0762	0.350383	0.175191	−	0.984534i	$-0.443946\pi$
$827$	10.0762	0.350383	0.175191	+	0.984534i	$0.443946\pi$
$828$	0	0
$829$	7.77750	0.270124	0.135062	−	0.990837i	$-0.456877\pi$
$829$	7.77750	0.270124	0.135062	+	0.990837i	$0.456877\pi$
$830$	0	0
$831$	47.3324	1.64194
$832$	0	0
$833$	−50.0432	−1.73389
$834$	0	0
$835$	20.2969	0.702404
$836$	0	0
$837$	−25.7674	−0.890651
$838$	0	0
$839$	−40.5765	−1.40086	−0.700428	−	0.713723i	$-0.747008\pi$
$839$	−40.5765	−1.40086	−0.700428	+	0.713723i	$0.747008\pi$
$840$	0	0
$841$	−19.1378	−0.659925
$842$	0	0
$843$	−66.9203	−2.30486
$844$	0	0
$845$	3.00154	0.103256
$846$	0	0
$847$	40.3048	1.38489
$848$	0	0
$849$	66.5412	2.28369
$850$	0	0
$851$	−2.10734	−0.0722388
$852$	0	0
$853$	−8.93899	−0.306065	−0.153033	−	0.988221i	$-0.548904\pi$
$853$	−8.93899	−0.306065	−0.153033	+	0.988221i	$0.548904\pi$
$854$	0	0
$855$	−4.65880	−0.159328
$856$	0	0
$857$	32.5955	1.11344	0.556721	−	0.830699i	$-0.312059\pi$
$857$	32.5955	1.11344	0.556721	+	0.830699i	$0.312059\pi$
$858$	0	0
$859$	0.247240	0.00843571	0.00421786	−	0.999991i	$-0.498657\pi$
$859$	0.247240	0.00843571	0.00421786	+	0.999991i	$0.498657\pi$
$860$	0	0
$861$	−87.3678	−2.97749
$862$	0	0
$863$	−4.91094	−0.167170	−0.0835852	−	0.996501i	$-0.526637\pi$
$863$	−4.91094	−0.167170	−0.0835852	+	0.996501i	$0.526637\pi$
$864$	0	0
$865$	16.1865	0.550358
$866$	0	0
$867$	−26.2863	−0.892729
$868$	0	0
$869$	8.98854	0.304915
$870$	0	0
$871$	−5.19868	−0.176150
$872$	0	0
$873$	−11.2783	−0.381712
$874$	0	0
$875$	−4.07774	−0.137853
$876$	0	0
$877$	−27.7665	−0.937607	−0.468804	−	0.883302i	$-0.655315\pi$
$877$	−27.7665	−0.937607	−0.468804	+	0.883302i	$0.655315\pi$
$878$	0	0
$879$	21.9418	0.740079
$880$	0	0
$881$	3.40430	0.114694	0.0573470	−	0.998354i	$-0.481736\pi$
$881$	3.40430	0.114694	0.0573470	+	0.998354i	$0.481736\pi$
$882$	0	0
$883$	35.9445	1.20963	0.604814	−	0.796367i	$-0.293247\pi$
$883$	35.9445	1.20963	0.604814	+	0.796367i	$0.293247\pi$
$884$	0	0
$885$	1.96771	0.0661439
$886$	0	0
$887$	36.6651	1.23109	0.615547	−	0.788100i	$-0.288935\pi$
$887$	36.6651	1.23109	0.615547	+	0.788100i	$0.288935\pi$
$888$	0	0
$889$	69.5192	2.33160
$890$	0	0
$891$	−5.86352	−0.196435
$892$	0	0
$893$	−10.4780	−0.350633
$894$	0	0
$895$	−3.87463	−0.129515
$896$	0	0
$897$	25.8305	0.862456
$898$	0	0
$899$	34.7498	1.15897
$900$	0	0
$901$	36.6141	1.21979
$902$	0	0
$903$	−21.4105	−0.712496
$904$	0	0
$905$	17.6434	0.586487
$906$	0	0
$907$	−16.0344	−0.532412	−0.266206	−	0.963916i	$-0.585770\pi$
$907$	−16.0344	−0.532412	−0.266206	+	0.963916i	$0.585770\pi$
$908$	0	0
$909$	−1.41934	−0.0470764
$910$	0	0
$911$	−21.5444	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$911$	−21.5444	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$912$	0	0
$913$	16.5021	0.546138
$914$	0	0
$915$	−36.7251	−1.21409
$916$	0	0
$917$	67.2221	2.21987
$918$	0	0
$919$	−14.1279	−0.466037	−0.233019	−	0.972472i	$-0.574860\pi$
$919$	−14.1279	−0.466037	−0.233019	+	0.972472i	$0.574860\pi$
$920$	0	0
$921$	56.7995	1.87161
$922$	0	0
$923$	−9.74049	−0.320612
$924$	0	0
$925$	−0.856462	−0.0281603
$926$	0	0
$927$	11.0143	0.361757
$928$	0	0
$929$	−20.0606	−0.658166	−0.329083	−	0.944301i	$-0.606740\pi$
$929$	−20.0606	−0.658166	−0.329083	+	0.944301i	$0.606740\pi$
$930$	0	0
$931$	−11.5387	−0.378165
$932$	0	0
$933$	−85.7420	−2.80707
$934$	0	0
$935$	−5.49062	−0.179563
$936$	0	0
$937$	17.8340	0.582610	0.291305	−	0.956630i	$-0.405911\pi$
$937$	17.8340	0.582610	0.291305	+	0.956630i	$0.405911\pi$
$938$	0	0
$939$	−57.6402	−1.88102
$940$	0	0
$941$	17.9930	0.586556	0.293278	−	0.956027i	$-0.405254\pi$
$941$	17.9930	0.586556	0.293278	+	0.956027i	$0.405254\pi$
$942$	0	0
$943$	−20.0879	−0.654152
$944$	0	0
$945$	9.49560	0.308892
$946$	0	0
$947$	−4.45198	−0.144670	−0.0723349	−	0.997380i	$-0.523045\pi$
$947$	−4.45198	−0.144670	−0.0723349	+	0.997380i	$0.523045\pi$
$948$	0	0
$949$	6.42618	0.208603
$950$	0	0
$951$	13.7359	0.445417
$952$	0	0
$953$	37.8089	1.22475	0.612375	−	0.790567i	$-0.290214\pi$
$953$	37.8089	1.22475	0.612375	+	0.790567i	$0.290214\pi$
$954$	0	0
$955$	24.0654	0.778738
$956$	0	0
$957$	−8.70603	−0.281426
$958$	0	0
$959$	−90.4508	−2.92081
$960$	0	0
$961$	91.4428	2.94977
$962$	0	0
$963$	15.1153	0.487083
$964$	0	0
$965$	0.501524	0.0161446
$966$	0	0
$967$	−22.7292	−0.730921	−0.365461	−	0.930827i	$-0.619088\pi$
$967$	−22.7292	−0.730921	−0.365461	+	0.930827i	$0.619088\pi$
$968$	0	0
$969$	−16.3479	−0.525169
$970$	0	0
$971$	30.9128	0.992038	0.496019	−	0.868312i	$-0.334795\pi$
$971$	30.9128	0.992038	0.496019	+	0.868312i	$0.334795\pi$
$972$	0	0
$973$	44.1929	1.41676
$974$	0	0
$975$	10.4980	0.336205
$976$	0	0
$977$	2.67852	0.0856934	0.0428467	−	0.999082i	$-0.486357\pi$
$977$	2.67852	0.0856934	0.0428467	+	0.999082i	$0.486357\pi$
$978$	0	0
$979$	−6.74258	−0.215494
$980$	0	0
$981$	1.22935	0.0392500
$982$	0	0
$983$	−47.0521	−1.50073	−0.750365	−	0.661024i	$-0.770122\pi$
$983$	−47.0521	−1.50073	−0.750365	+	0.661024i	$0.770122\pi$
$984$	0	0
$985$	−18.5551	−0.591215
$986$	0	0
$987$	93.5617	2.97810
$988$	0	0
$989$	−4.92276	−0.156535
$990$	0	0
$991$	−44.0919	−1.40062	−0.700312	−	0.713837i	$-0.746956\pi$
$991$	−44.0919	−1.40062	−0.700312	+	0.713837i	$0.746956\pi$
$992$	0	0
$993$	4.95947	0.157384
$994$	0	0
$995$	−9.99392	−0.316829
$996$	0	0
$997$	34.2940	1.08610	0.543051	−	0.839700i	$-0.317269\pi$
$997$	34.2940	1.08610	0.543051	+	0.839700i	$0.317269\pi$
$998$	0	0
$999$	1.99440	0.0630999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.4	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.4	✓	29	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)

\(f(q)\)	\(=\)	\(q-2.62437 q^{3} +1.00000 q^{5} -4.07774 q^{7} +3.88732 q^{9} +O(q^{10})\)	\(q-2.62437 q^{3} +1.00000 q^{5} -4.07774 q^{7} +3.88732 q^{9} +1.05635 q^{11} -4.00019 q^{13} -2.62437 q^{15} -5.19771 q^{17} -1.19846 q^{19} +10.7015 q^{21} +2.46052 q^{23} +1.00000 q^{25} -2.32865 q^{27} +3.14041 q^{29} +11.0654 q^{31} -2.77226 q^{33} -4.07774 q^{35} -0.856462 q^{37} +10.4980 q^{39} -8.16409 q^{41} -2.00070 q^{43} +3.88732 q^{45} +8.74288 q^{47} +9.62792 q^{49} +13.6407 q^{51} -7.04428 q^{53} +1.05635 q^{55} +3.14521 q^{57} -0.749784 q^{59} +13.9939 q^{61} -15.8514 q^{63} -4.00019 q^{65} +1.29961 q^{67} -6.45732 q^{69} +2.43501 q^{71} -1.60647 q^{73} -2.62437 q^{75} -4.30753 q^{77} +8.50903 q^{79} -5.55072 q^{81} +15.6217 q^{83} -5.19771 q^{85} -8.24159 q^{87} -6.38288 q^{89} +16.3117 q^{91} -29.0397 q^{93} -1.19846 q^{95} -2.90130 q^{97} +4.10638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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