LMFDB - Embedded newform 920.4.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,4,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.2817572053$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 $ x^6 - 2x^5 - 81x^4 + 161x^3 + 1520x^2 - 3915*x + 588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.160327$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.160327 q^{3} -5.00000 q^{5} -2.26267 q^{7} -26.9743 q^{9} -8.80426 q^{11} +37.4404 q^{13} -0.801636 q^{15} +71.0912 q^{17} +106.599 q^{19} -0.362767 q^{21} -23.0000 q^{23} +25.0000 q^{25} -8.65355 q^{27} -10.8619 q^{29} +42.2033 q^{31} -1.41156 q^{33} +11.3133 q^{35} -140.769 q^{37} +6.00272 q^{39} +1.84812 q^{41} -503.969 q^{43} +134.871 q^{45} -219.338 q^{47} -337.880 q^{49} +11.3979 q^{51} -59.4388 q^{53} +44.0213 q^{55} +17.0906 q^{57} -91.0125 q^{59} -93.1843 q^{61} +61.0338 q^{63} -187.202 q^{65} -940.265 q^{67} -3.68753 q^{69} -178.289 q^{71} -47.2604 q^{73} +4.00818 q^{75} +19.9211 q^{77} +1082.23 q^{79} +726.919 q^{81} +484.557 q^{83} -355.456 q^{85} -1.74145 q^{87} -392.280 q^{89} -84.7151 q^{91} +6.76634 q^{93} -532.993 q^{95} -1650.20 q^{97} +237.489 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.160327	0.0308550	0.0154275	−	0.999881i	$-0.495089\pi$
$3$	0.160327	0.0308550	0.0154275	+	0.999881i	$0.495089\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−2.26267	−0.122172	−0.0610862	−	0.998132i	$-0.519456\pi$
$7$	−2.26267	−0.122172	−0.0610862	+	0.998132i	$0.519456\pi$
$8$	0	0
$9$	−26.9743	−0.999048
$10$	0	0
$11$	−8.80426	−0.241326	−0.120663	−	0.992694i	$-0.538502\pi$
$11$	−8.80426	−0.241326	−0.120663	+	0.992694i	$0.538502\pi$
$12$	0	0
$13$	37.4404	0.798777	0.399389	−	0.916782i	$-0.369222\pi$
$13$	37.4404	0.798777	0.399389	+	0.916782i	$0.369222\pi$
$14$	0	0
$15$	−0.801636	−0.0137988
$16$	0	0
$17$	71.0912	1.01424	0.507122	−	0.861874i	$-0.330709\pi$
$17$	71.0912	1.01424	0.507122	+	0.861874i	$0.330709\pi$
$18$	0	0
$19$	106.599	1.28713	0.643563	−	0.765393i	$-0.277456\pi$
$19$	106.599	1.28713	0.643563	+	0.765393i	$0.277456\pi$
$20$	0	0
$21$	−0.362767	−0.00376963
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−8.65355	−0.0616806
$28$	0	0
$29$	−10.8619	−0.0695516	−0.0347758	−	0.999395i	$-0.511072\pi$
$29$	−10.8619	−0.0695516	−0.0347758	+	0.999395i	$0.511072\pi$
$30$	0	0
$31$	42.2033	0.244514	0.122257	−	0.992498i	$-0.460987\pi$
$31$	42.2033	0.244514	0.122257	+	0.992498i	$0.460987\pi$
$32$	0	0
$33$	−1.41156	−0.00744611
$34$	0	0
$35$	11.3133	0.0546372
$36$	0	0
$37$	−140.769	−0.625468	−0.312734	−	0.949841i	$-0.601245\pi$
$37$	−140.769	−0.625468	−0.312734	+	0.949841i	$0.601245\pi$
$38$	0	0
$39$	6.00272	0.0246463
$40$	0	0
$41$	1.84812	0.00703969	0.00351985	−	0.999994i	$-0.498880\pi$
$41$	1.84812	0.00703969	0.00351985	+	0.999994i	$0.498880\pi$
$42$	0	0
$43$	−503.969	−1.78731	−0.893657	−	0.448751i	$-0.851869\pi$
$43$	−503.969	−1.78731	−0.893657	+	0.448751i	$0.851869\pi$
$44$	0	0
$45$	134.871	0.446788
$46$	0	0
$47$	−219.338	−0.680719	−0.340359	−	0.940295i	$-0.610549\pi$
$47$	−219.338	−0.680719	−0.340359	+	0.940295i	$0.610549\pi$
$48$	0	0
$49$	−337.880	−0.985074
$50$	0	0
$51$	11.3979	0.0312945
$52$	0	0
$53$	−59.4388	−0.154048	−0.0770240	−	0.997029i	$-0.524542\pi$
$53$	−59.4388	−0.154048	−0.0770240	+	0.997029i	$0.524542\pi$
$54$	0	0
$55$	44.0213	0.107924
$56$	0	0
$57$	17.0906	0.0397142
$58$	0	0
$59$	−91.0125	−0.200828	−0.100414	−	0.994946i	$-0.532017\pi$
$59$	−91.0125	−0.200828	−0.100414	+	0.994946i	$0.532017\pi$
$60$	0	0
$61$	−93.1843	−0.195591	−0.0977953	−	0.995207i	$-0.531179\pi$
$61$	−93.1843	−0.195591	−0.0977953	+	0.995207i	$0.531179\pi$
$62$	0	0
$63$	61.0338	0.122056
$64$	0	0
$65$	−187.202	−0.357224
$66$	0	0
$67$	−940.265	−1.71450	−0.857251	−	0.514899i	$-0.827830\pi$
$67$	−940.265	−1.71450	−0.857251	+	0.514899i	$0.827830\pi$
$68$	0	0
$69$	−3.68753	−0.00643371
$70$	0	0
$71$	−178.289	−0.298015	−0.149007	−	0.988836i	$-0.547608\pi$
$71$	−178.289	−0.298015	−0.149007	+	0.988836i	$0.547608\pi$
$72$	0	0
$73$	−47.2604	−0.0757728	−0.0378864	−	0.999282i	$-0.512062\pi$
$73$	−47.2604	−0.0757728	−0.0378864	+	0.999282i	$0.512062\pi$
$74$	0	0
$75$	4.00818	0.00617100
$76$	0	0
$77$	19.9211	0.0294834
$78$	0	0
$79$	1082.23	1.54126	0.770632	−	0.637280i	$-0.219941\pi$
$79$	1082.23	1.54126	0.770632	+	0.637280i	$0.219941\pi$
$80$	0	0
$81$	726.919	0.997145
$82$	0	0
$83$	484.557	0.640808	0.320404	−	0.947281i	$-0.396181\pi$
$83$	484.557	0.640808	0.320404	+	0.947281i	$0.396181\pi$
$84$	0	0
$85$	−355.456	−0.453584
$86$	0	0
$87$	−1.74145	−0.00214601
$88$	0	0
$89$	−392.280	−0.467209	−0.233604	−	0.972332i	$-0.575052\pi$
$89$	−392.280	−0.467209	−0.233604	+	0.972332i	$0.575052\pi$
$90$	0	0
$91$	−84.7151	−0.0975886
$92$	0	0
$93$	6.76634	0.00754448
$94$	0	0
$95$	−532.993	−0.575620
$96$	0	0
$97$	−1650.20	−1.72734	−0.863672	−	0.504054i	$-0.831841\pi$
$97$	−1650.20	−1.72734	−0.863672	+	0.504054i	$0.831841\pi$
$98$	0	0
$99$	237.489	0.241096
$100$	0	0
$101$	−1209.95	−1.19202	−0.596011	−	0.802977i	$-0.703248\pi$
$101$	−1209.95	−1.19202	−0.596011	+	0.802977i	$0.703248\pi$
$102$	0	0
$103$	−1214.33	−1.16167	−0.580834	−	0.814022i	$-0.697273\pi$
$103$	−1214.33	−1.16167	−0.580834	+	0.814022i	$0.697273\pi$
$104$	0	0
$105$	1.81383	0.00168583
$106$	0	0
$107$	881.150	0.796112	0.398056	−	0.917361i	$-0.369685\pi$
$107$	881.150	0.796112	0.398056	+	0.917361i	$0.369685\pi$
$108$	0	0
$109$	−1400.89	−1.23102	−0.615509	−	0.788130i	$-0.711050\pi$
$109$	−1400.89	−1.23102	−0.615509	+	0.788130i	$0.711050\pi$
$110$	0	0
$111$	−22.5691	−0.0192988
$112$	0	0
$113$	2303.58	1.91772	0.958862	−	0.283873i	$-0.0916196\pi$
$113$	2303.58	1.91772	0.958862	+	0.283873i	$0.0916196\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−1009.93	−0.798017
$118$	0	0
$119$	−160.856	−0.123913
$120$	0	0
$121$	−1253.48	−0.941762
$122$	0	0
$123$	0.296304	0.000217210 0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	29.7959	0.0208186	0.0104093	−	0.999946i	$-0.496687\pi$
$127$	29.7959	0.0208186	0.0104093	+	0.999946i	$0.496687\pi$
$128$	0	0
$129$	−80.7999	−0.0551476
$130$	0	0
$131$	−944.060	−0.629640	−0.314820	−	0.949151i	$-0.601944\pi$
$131$	−944.060	−0.629640	−0.314820	+	0.949151i	$0.601944\pi$
$132$	0	0
$133$	−241.197	−0.157251
$134$	0	0
$135$	43.2677	0.0275844
$136$	0	0
$137$	−1545.96	−0.964091	−0.482046	−	0.876146i	$-0.660106\pi$
$137$	−1545.96	−0.964091	−0.482046	+	0.876146i	$0.660106\pi$
$138$	0	0
$139$	−464.196	−0.283256	−0.141628	−	0.989920i	$-0.545234\pi$
$139$	−464.196	−0.283256	−0.141628	+	0.989920i	$0.545234\pi$
$140$	0	0
$141$	−35.1659	−0.0210036
$142$	0	0
$143$	−329.635	−0.192766
$144$	0	0
$145$	54.3093	0.0311044
$146$	0	0
$147$	−54.1714	−0.0303944
$148$	0	0
$149$	−2401.34	−1.32030	−0.660152	−	0.751132i	$-0.729508\pi$
$149$	−2401.34	−1.32030	−0.660152	+	0.751132i	$0.729508\pi$
$150$	0	0
$151$	2669.19	1.43851	0.719256	−	0.694745i	$-0.244483\pi$
$151$	2669.19	1.43851	0.719256	+	0.694745i	$0.244483\pi$
$152$	0	0
$153$	−1917.64	−1.01328
$154$	0	0
$155$	−211.017	−0.109350
$156$	0	0
$157$	1946.79	0.989622	0.494811	−	0.869001i	$-0.335237\pi$
$157$	1946.79	0.989622	0.494811	+	0.869001i	$0.335237\pi$
$158$	0	0
$159$	−9.52965	−0.00475315
$160$	0	0
$161$	52.0413	0.0254747
$162$	0	0
$163$	2336.46	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$163$	2336.46	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$164$	0	0
$165$	7.05782	0.00333000
$166$	0	0
$167$	−1787.58	−0.828307	−0.414153	−	0.910207i	$-0.635922\pi$
$167$	−1787.58	−0.828307	−0.414153	+	0.910207i	$0.635922\pi$
$168$	0	0
$169$	−795.215	−0.361955
$170$	0	0
$171$	−2875.42	−1.28590
$172$	0	0
$173$	−2033.68	−0.893744	−0.446872	−	0.894598i	$-0.647462\pi$
$173$	−2033.68	−0.893744	−0.446872	+	0.894598i	$0.647462\pi$
$174$	0	0
$175$	−56.5666	−0.0244345
$176$	0	0
$177$	−14.5918	−0.00619653
$178$	0	0
$179$	3860.15	1.61185	0.805926	−	0.592016i	$-0.201668\pi$
$179$	3860.15	1.61185	0.805926	+	0.592016i	$0.201668\pi$
$180$	0	0
$181$	−2735.90	−1.12353	−0.561763	−	0.827298i	$-0.689877\pi$
$181$	−2735.90	−1.12353	−0.561763	+	0.827298i	$0.689877\pi$
$182$	0	0
$183$	−14.9400	−0.00603494
$184$	0	0
$185$	703.846	0.279718
$186$	0	0
$187$	−625.906	−0.244763
$188$	0	0
$189$	19.5801	0.00753567
$190$	0	0
$191$	−2818.53	−1.06776	−0.533878	−	0.845561i	$-0.679266\pi$
$191$	−2818.53	−1.06776	−0.533878	+	0.845561i	$0.679266\pi$
$192$	0	0
$193$	1493.07	0.556858	0.278429	−	0.960457i	$-0.410186\pi$
$193$	1493.07	0.556858	0.278429	+	0.960457i	$0.410186\pi$
$194$	0	0
$195$	−30.0136	−0.0110221
$196$	0	0
$197$	2748.53	0.994033	0.497017	−	0.867741i	$-0.334429\pi$
$197$	2748.53	0.994033	0.497017	+	0.867741i	$0.334429\pi$
$198$	0	0
$199$	−3400.87	−1.21146	−0.605732	−	0.795669i	$-0.707120\pi$
$199$	−3400.87	−1.21146	−0.605732	+	0.795669i	$0.707120\pi$
$200$	0	0
$201$	−150.750	−0.0529009
$202$	0	0
$203$	24.5767	0.00849729
$204$	0	0
$205$	−9.24059	−0.00314825
$206$	0	0
$207$	620.409	0.208316
$208$	0	0
$209$	−938.522	−0.310617
$210$	0	0
$211$	−1348.57	−0.439998	−0.219999	−	0.975500i	$-0.570605\pi$
$211$	−1348.57	−0.439998	−0.219999	+	0.975500i	$0.570605\pi$
$212$	0	0
$213$	−28.5846	−0.00919524
$214$	0	0
$215$	2519.84	0.799311
$216$	0	0
$217$	−95.4920	−0.0298729
$218$	0	0
$219$	−7.57713	−0.00233797
$220$	0	0
$221$	2661.68	0.810155
$222$	0	0
$223$	−196.177	−0.0589102	−0.0294551	−	0.999566i	$-0.509377\pi$
$223$	−196.177	−0.0589102	−0.0294551	+	0.999566i	$0.509377\pi$
$224$	0	0
$225$	−674.357	−0.199810
$226$	0	0
$227$	5337.12	1.56052	0.780258	−	0.625457i	$-0.215088\pi$
$227$	5337.12	1.56052	0.780258	+	0.625457i	$0.215088\pi$
$228$	0	0
$229$	−499.327	−0.144089	−0.0720447	−	0.997401i	$-0.522952\pi$
$229$	−499.327	−0.144089	−0.0720447	+	0.997401i	$0.522952\pi$
$230$	0	0
$231$	3.19390	0.000909709 0
$232$	0	0
$233$	−4591.49	−1.29098	−0.645490	−	0.763769i	$-0.723347\pi$
$233$	−4591.49	−1.29098	−0.645490	+	0.763769i	$0.723347\pi$
$234$	0	0
$235$	1096.69	0.304427
$236$	0	0
$237$	173.510	0.0475557
$238$	0	0
$239$	−800.167	−0.216563	−0.108281	−	0.994120i	$-0.534535\pi$
$239$	−800.167	−0.216563	−0.108281	+	0.994120i	$0.534535\pi$
$240$	0	0
$241$	−4311.30	−1.15234	−0.576172	−	0.817328i	$-0.695454\pi$
$241$	−4311.30	−1.15234	−0.576172	+	0.817328i	$0.695454\pi$
$242$	0	0
$243$	350.191	0.0924475
$244$	0	0
$245$	1689.40	0.440538
$246$	0	0
$247$	3991.09	1.02813
$248$	0	0
$249$	77.6877	0.0197721
$250$	0	0
$251$	−1791.71	−0.450564	−0.225282	−	0.974294i	$-0.572330\pi$
$251$	−1791.71	−0.450564	−0.225282	+	0.974294i	$0.572330\pi$
$252$	0	0
$253$	202.498	0.0503199
$254$	0	0
$255$	−56.9893	−0.0139953
$256$	0	0
$257$	−6620.73	−1.60697	−0.803483	−	0.595328i	$-0.797022\pi$
$257$	−6620.73	−1.60697	−0.803483	+	0.595328i	$0.797022\pi$
$258$	0	0
$259$	318.513	0.0764149
$260$	0	0
$261$	292.991	0.0694854
$262$	0	0
$263$	8208.98	1.92467	0.962333	−	0.271873i	$-0.0876432\pi$
$263$	8208.98	1.92467	0.962333	+	0.271873i	$0.0876432\pi$
$264$	0	0
$265$	297.194	0.0688923
$266$	0	0
$267$	−62.8931	−0.0144157
$268$	0	0
$269$	−1976.84	−0.448068	−0.224034	−	0.974581i	$-0.571923\pi$
$269$	−1976.84	−0.448068	−0.224034	+	0.974581i	$0.571923\pi$
$270$	0	0
$271$	4413.16	0.989226	0.494613	−	0.869113i	$-0.335310\pi$
$271$	4413.16	0.989226	0.494613	+	0.869113i	$0.335310\pi$
$272$	0	0
$273$	−13.5821	−0.00301109
$274$	0	0
$275$	−220.107	−0.0482652
$276$	0	0
$277$	−4558.54	−0.988794	−0.494397	−	0.869236i	$-0.664611\pi$
$277$	−4558.54	−0.988794	−0.494397	+	0.869236i	$0.664611\pi$
$278$	0	0
$279$	−1138.40	−0.244281
$280$	0	0
$281$	−3221.73	−0.683959	−0.341980	−	0.939707i	$-0.611097\pi$
$281$	−3221.73	−0.683959	−0.341980	+	0.939707i	$0.611097\pi$
$282$	0	0
$283$	−4111.86	−0.863691	−0.431845	−	0.901948i	$-0.642137\pi$
$283$	−4111.86	−0.863691	−0.431845	+	0.901948i	$0.642137\pi$
$284$	0	0
$285$	−85.4532	−0.0177607
$286$	0	0
$287$	−4.18167	−0.000860057 0
$288$	0	0
$289$	140.959	0.0286910
$290$	0	0
$291$	−264.572	−0.0532972
$292$	0	0
$293$	−1558.54	−0.310754	−0.155377	−	0.987855i	$-0.549659\pi$
$293$	−1558.54	−0.310754	−0.155377	+	0.987855i	$0.549659\pi$
$294$	0	0
$295$	455.063	0.0898128
$296$	0	0
$297$	76.1881	0.0148851
$298$	0	0
$299$	−861.130	−0.166557
$300$	0	0
$301$	1140.31	0.218360
$302$	0	0
$303$	−193.987	−0.0367798
$304$	0	0
$305$	465.921	0.0874707
$306$	0	0
$307$	−1944.80	−0.361550	−0.180775	−	0.983525i	$-0.557861\pi$
$307$	−1944.80	−0.361550	−0.180775	+	0.983525i	$0.557861\pi$
$308$	0	0
$309$	−194.691	−0.0358432
$310$	0	0
$311$	1359.73	0.247921	0.123960	−	0.992287i	$-0.460440\pi$
$311$	1359.73	0.247921	0.123960	+	0.992287i	$0.460440\pi$
$312$	0	0
$313$	−730.171	−0.131858	−0.0659292	−	0.997824i	$-0.521001\pi$
$313$	−730.171	−0.131858	−0.0659292	+	0.997824i	$0.521001\pi$
$314$	0	0
$315$	−305.169	−0.0545852
$316$	0	0
$317$	2298.93	0.407321	0.203661	−	0.979042i	$-0.434716\pi$
$317$	2298.93	0.407321	0.203661	+	0.979042i	$0.434716\pi$
$318$	0	0
$319$	95.6306	0.0167846
$320$	0	0
$321$	141.272	0.0245640
$322$	0	0
$323$	7578.22	1.30546
$324$	0	0
$325$	936.010	0.159755
$326$	0	0
$327$	−224.601	−0.0379831
$328$	0	0
$329$	496.289	0.0831650
$330$	0	0
$331$	4043.94	0.671526	0.335763	−	0.941947i	$-0.391006\pi$
$331$	4043.94	0.671526	0.335763	+	0.941947i	$0.391006\pi$
$332$	0	0
$333$	3797.15	0.624872
$334$	0	0
$335$	4701.32	0.766749
$336$	0	0
$337$	−11725.3	−1.89530	−0.947650	−	0.319311i	$-0.896549\pi$
$337$	−11725.3	−1.89530	−0.947650	+	0.319311i	$0.896549\pi$
$338$	0	0
$339$	369.327	0.0591713
$340$	0	0
$341$	−371.569	−0.0590076
$342$	0	0
$343$	1540.60	0.242521
$344$	0	0
$345$	18.4376	0.00287724
$346$	0	0
$347$	−1835.72	−0.283997	−0.141998	−	0.989867i	$-0.545353\pi$
$347$	−1835.72	−0.283997	−0.141998	+	0.989867i	$0.545353\pi$
$348$	0	0
$349$	8404.26	1.28902	0.644512	−	0.764594i	$-0.277060\pi$
$349$	8404.26	1.28902	0.644512	+	0.764594i	$0.277060\pi$
$350$	0	0
$351$	−323.992	−0.0492691
$352$	0	0
$353$	7903.48	1.19167	0.595835	−	0.803107i	$-0.296821\pi$
$353$	7903.48	1.19167	0.595835	+	0.803107i	$0.296821\pi$
$354$	0	0
$355$	891.447	0.133276
$356$	0	0
$357$	−25.7895	−0.00382332
$358$	0	0
$359$	7864.66	1.15621	0.578107	−	0.815961i	$-0.303792\pi$
$359$	7864.66	1.15621	0.578107	+	0.815961i	$0.303792\pi$
$360$	0	0
$361$	4504.25	0.656692
$362$	0	0
$363$	−200.968	−0.0290581
$364$	0	0
$365$	236.302	0.0338866
$366$	0	0
$367$	−5339.97	−0.759521	−0.379760	−	0.925085i	$-0.623994\pi$
$367$	−5339.97	−0.759521	−0.379760	+	0.925085i	$0.623994\pi$
$368$	0	0
$369$	−49.8517	−0.00703299
$370$	0	0
$371$	134.490	0.0188204
$372$	0	0
$373$	9071.99	1.25933	0.629665	−	0.776867i	$-0.283192\pi$
$373$	9071.99	1.25933	0.629665	+	0.776867i	$0.283192\pi$
$374$	0	0
$375$	−20.0409	−0.00275975
$376$	0	0
$377$	−406.672	−0.0555562
$378$	0	0
$379$	2133.10	0.289103	0.144552	−	0.989497i	$-0.453826\pi$
$379$	2133.10	0.289103	0.144552	+	0.989497i	$0.453826\pi$
$380$	0	0
$381$	4.77709	0.000642357 0
$382$	0	0
$383$	11184.4	1.49216	0.746081	−	0.665855i	$-0.231933\pi$
$383$	11184.4	1.49216	0.746081	+	0.665855i	$0.231933\pi$
$384$	0	0
$385$	−99.6055	−0.0131854
$386$	0	0
$387$	13594.2	1.78561
$388$	0	0
$389$	9802.21	1.27761	0.638807	−	0.769367i	$-0.279428\pi$
$389$	9802.21	1.27761	0.638807	+	0.769367i	$0.279428\pi$
$390$	0	0
$391$	−1635.10	−0.211485
$392$	0	0
$393$	−151.358	−0.0194275
$394$	0	0
$395$	−5411.13	−0.689274
$396$	0	0
$397$	9500.17	1.20101	0.600504	−	0.799622i	$-0.294967\pi$
$397$	9500.17	1.20101	0.600504	+	0.799622i	$0.294967\pi$
$398$	0	0
$399$	−38.6704	−0.00485199
$400$	0	0
$401$	−8889.43	−1.10703	−0.553513	−	0.832841i	$-0.686713\pi$
$401$	−8889.43	−1.10703	−0.553513	+	0.832841i	$0.686713\pi$
$402$	0	0
$403$	1580.11	0.195312
$404$	0	0
$405$	−3634.59	−0.445937
$406$	0	0
$407$	1239.37	0.150942
$408$	0	0
$409$	−14456.7	−1.74777	−0.873886	−	0.486131i	$-0.838408\pi$
$409$	−14456.7	−1.74777	−0.873886	+	0.486131i	$0.838408\pi$
$410$	0	0
$411$	−247.860	−0.0297470
$412$	0	0
$413$	205.931	0.0245356
$414$	0	0
$415$	−2422.78	−0.286578
$416$	0	0
$417$	−74.4233	−0.00873987
$418$	0	0
$419$	3558.06	0.414851	0.207426	−	0.978251i	$-0.433491\pi$
$419$	3558.06	0.414851	0.207426	+	0.978251i	$0.433491\pi$
$420$	0	0
$421$	10410.1	1.20512	0.602562	−	0.798072i	$-0.294146\pi$
$421$	10410.1	1.20512	0.602562	+	0.798072i	$0.294146\pi$
$422$	0	0
$423$	5916.49	0.680070
$424$	0	0
$425$	1777.28	0.202849
$426$	0	0
$427$	210.845	0.0238958
$428$	0	0
$429$	−52.8495	−0.00594778
$430$	0	0
$431$	−10212.3	−1.14132	−0.570658	−	0.821188i	$-0.693312\pi$
$431$	−10212.3	−1.14132	−0.570658	+	0.821188i	$0.693312\pi$
$432$	0	0
$433$	13756.0	1.52672	0.763359	−	0.645975i	$-0.223549\pi$
$433$	13756.0	1.52672	0.763359	+	0.645975i	$0.223549\pi$
$434$	0	0
$435$	8.70726	0.000959726 0
$436$	0	0
$437$	−2451.77	−0.268384
$438$	0	0
$439$	−3625.86	−0.394198	−0.197099	−	0.980384i	$-0.563152\pi$
$439$	−3625.86	−0.394198	−0.197099	+	0.980384i	$0.563152\pi$
$440$	0	0
$441$	9114.08	0.984136
$442$	0	0
$443$	−2133.43	−0.228809	−0.114405	−	0.993434i	$-0.536496\pi$
$443$	−2133.43	−0.228809	−0.114405	+	0.993434i	$0.536496\pi$
$444$	0	0
$445$	1961.40	0.208942
$446$	0	0
$447$	−385.000	−0.0407379
$448$	0	0
$449$	−1351.97	−0.142101	−0.0710504	−	0.997473i	$-0.522635\pi$
$449$	−1351.97	−0.142101	−0.0710504	+	0.997473i	$0.522635\pi$
$450$	0	0
$451$	−16.2713	−0.00169886
$452$	0	0
$453$	427.944	0.0443853
$454$	0	0
$455$	423.576	0.0436429
$456$	0	0
$457$	−11704.0	−1.19801	−0.599006	−	0.800745i	$-0.704437\pi$
$457$	−11704.0	−1.19801	−0.599006	+	0.800745i	$0.704437\pi$
$458$	0	0
$459$	−615.191	−0.0625592
$460$	0	0
$461$	−15627.8	−1.57887	−0.789434	−	0.613835i	$-0.789626\pi$
$461$	−15627.8	−1.57887	−0.789434	+	0.613835i	$0.789626\pi$
$462$	0	0
$463$	−1002.05	−0.100581	−0.0502907	−	0.998735i	$-0.516015\pi$
$463$	−1002.05	−0.100581	−0.0502907	+	0.998735i	$0.516015\pi$
$464$	0	0
$465$	−33.8317	−0.00337399
$466$	0	0
$467$	3247.30	0.321771	0.160886	−	0.986973i	$-0.448565\pi$
$467$	3247.30	0.321771	0.160886	+	0.986973i	$0.448565\pi$
$468$	0	0
$469$	2127.50	0.209465
$470$	0	0
$471$	312.123	0.0305348
$472$	0	0
$473$	4437.07	0.431325
$474$	0	0
$475$	2664.96	0.257425
$476$	0	0
$477$	1603.32	0.153901
$478$	0	0
$479$	16235.1	1.54865	0.774324	−	0.632789i	$-0.218090\pi$
$479$	16235.1	1.54865	0.774324	+	0.632789i	$0.218090\pi$
$480$	0	0
$481$	−5270.45	−0.499609
$482$	0	0
$483$	8.34364	0.000786022 0
$484$	0	0
$485$	8251.00	0.772492
$486$	0	0
$487$	−4937.48	−0.459422	−0.229711	−	0.973259i	$-0.573778\pi$
$487$	−4937.48	−0.459422	−0.229711	+	0.973259i	$0.573778\pi$
$488$	0	0
$489$	374.598	0.0346419
$490$	0	0
$491$	−4804.96	−0.441640	−0.220820	−	0.975315i	$-0.570873\pi$
$491$	−4804.96	−0.441640	−0.220820	+	0.975315i	$0.570873\pi$
$492$	0	0
$493$	−772.182	−0.0705423
$494$	0	0
$495$	−1187.44	−0.107821
$496$	0	0
$497$	403.409	0.0364092
$498$	0	0
$499$	1571.95	0.141022	0.0705111	−	0.997511i	$-0.477537\pi$
$499$	1571.95	0.141022	0.0705111	+	0.997511i	$0.477537\pi$
$500$	0	0
$501$	−286.598	−0.0255574
$502$	0	0
$503$	10608.7	0.940391	0.470195	−	0.882562i	$-0.344183\pi$
$503$	10608.7	0.940391	0.470195	+	0.882562i	$0.344183\pi$
$504$	0	0
$505$	6049.73	0.533088
$506$	0	0
$507$	−127.495	−0.0111681
$508$	0	0
$509$	−7020.51	−0.611353	−0.305677	−	0.952135i	$-0.598883\pi$
$509$	−7020.51	−0.611353	−0.305677	+	0.952135i	$0.598883\pi$
$510$	0	0
$511$	106.934	0.00925734
$512$	0	0
$513$	−922.456	−0.0793907
$514$	0	0
$515$	6071.66	0.519513
$516$	0	0
$517$	1931.11	0.164275
$518$	0	0
$519$	−326.054	−0.0275765
$520$	0	0
$521$	−8671.92	−0.729220	−0.364610	−	0.931160i	$-0.618798\pi$
$521$	−8671.92	−0.729220	−0.364610	+	0.931160i	$0.618798\pi$
$522$	0	0
$523$	11590.2	0.969033	0.484516	−	0.874782i	$-0.338996\pi$
$523$	11590.2	0.969033	0.484516	+	0.874782i	$0.338996\pi$
$524$	0	0
$525$	−9.06917	−0.000753926 0
$526$	0	0
$527$	3000.28	0.247997
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	2455.00	0.200636
$532$	0	0
$533$	69.1943	0.00562315
$534$	0	0
$535$	−4405.75	−0.356032
$536$	0	0
$537$	618.888	0.0497337
$538$	0	0
$539$	2974.79	0.237724
$540$	0	0
$541$	−1473.20	−0.117076	−0.0585379	−	0.998285i	$-0.518644\pi$
$541$	−1473.20	−0.117076	−0.0585379	+	0.998285i	$0.518644\pi$
$542$	0	0
$543$	−438.640	−0.0346664
$544$	0	0
$545$	7004.46	0.550528
$546$	0	0
$547$	3796.89	0.296789	0.148394	−	0.988928i	$-0.452589\pi$
$547$	3796.89	0.296789	0.148394	+	0.988928i	$0.452589\pi$
$548$	0	0
$549$	2513.58	0.195404
$550$	0	0
$551$	−1157.86	−0.0895216
$552$	0	0
$553$	−2448.71	−0.188300
$554$	0	0
$555$	112.846	0.00863068
$556$	0	0
$557$	9329.90	0.709732	0.354866	−	0.934917i	$-0.384527\pi$
$557$	9329.90	0.709732	0.354866	+	0.934917i	$0.384527\pi$
$558$	0	0
$559$	−18868.8	−1.42767
$560$	0	0
$561$	−100.350	−0.00755217
$562$	0	0
$563$	2267.79	0.169762	0.0848811	−	0.996391i	$-0.472949\pi$
$563$	2267.79	0.169762	0.0848811	+	0.996391i	$0.472949\pi$
$564$	0	0
$565$	−11517.9	−0.857632
$566$	0	0
$567$	−1644.77	−0.121824
$568$	0	0
$569$	16420.4	1.20980	0.604901	−	0.796301i	$-0.293213\pi$
$569$	16420.4	1.20980	0.604901	+	0.796301i	$0.293213\pi$
$570$	0	0
$571$	−16864.2	−1.23598	−0.617992	−	0.786185i	$-0.712053\pi$
$571$	−16864.2	−1.23598	−0.617992	+	0.786185i	$0.712053\pi$
$572$	0	0
$573$	−451.887	−0.0329456
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−8766.94	−0.632535	−0.316267	−	0.948670i	$-0.602430\pi$
$577$	−8766.94	−0.632535	−0.316267	+	0.948670i	$0.602430\pi$
$578$	0	0
$579$	239.380	0.0171819
$580$	0	0
$581$	−1096.39	−0.0782890
$582$	0	0
$583$	523.315	0.0371758
$584$	0	0
$585$	5049.64	0.356884
$586$	0	0
$587$	−15470.3	−1.08778	−0.543891	−	0.839156i	$-0.683049\pi$
$587$	−15470.3	−1.08778	−0.543891	+	0.839156i	$0.683049\pi$
$588$	0	0
$589$	4498.81	0.314720
$590$	0	0
$591$	440.664	0.0306709
$592$	0	0
$593$	2230.18	0.154439	0.0772197	−	0.997014i	$-0.475396\pi$
$593$	2230.18	0.154439	0.0772197	+	0.997014i	$0.475396\pi$
$594$	0	0
$595$	804.278	0.0554154
$596$	0	0
$597$	−545.252	−0.0373797
$598$	0	0
$599$	−7297.37	−0.497767	−0.248884	−	0.968533i	$-0.580064\pi$
$599$	−7297.37	−0.497767	−0.248884	+	0.968533i	$0.580064\pi$
$600$	0	0
$601$	−8275.74	−0.561688	−0.280844	−	0.959753i	$-0.590614\pi$
$601$	−8275.74	−0.561688	−0.280844	+	0.959753i	$0.590614\pi$
$602$	0	0
$603$	25363.0	1.71287
$604$	0	0
$605$	6267.42	0.421169
$606$	0	0
$607$	5278.34	0.352951	0.176475	−	0.984305i	$-0.443530\pi$
$607$	5278.34	0.352951	0.176475	+	0.984305i	$0.443530\pi$
$608$	0	0
$609$	3.94032	0.000262184 0
$610$	0	0
$611$	−8212.11	−0.543742
$612$	0	0
$613$	5420.18	0.357127	0.178564	−	0.983928i	$-0.442855\pi$
$613$	5420.18	0.357127	0.178564	+	0.983928i	$0.442855\pi$
$614$	0	0
$615$	−1.48152	−9.71391e−5 0
$616$	0	0
$617$	−20142.5	−1.31428	−0.657138	−	0.753770i	$-0.728233\pi$
$617$	−20142.5	−1.31428	−0.657138	+	0.753770i	$0.728233\pi$
$618$	0	0
$619$	−9432.84	−0.612500	−0.306250	−	0.951951i	$-0.599074\pi$
$619$	−9432.84	−0.612500	−0.306250	+	0.951951i	$0.599074\pi$
$620$	0	0
$621$	199.032	0.0128613
$622$	0	0
$623$	887.598	0.0570800
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−150.471	−0.00958408
$628$	0	0
$629$	−10007.4	−0.634377
$630$	0	0
$631$	−28702.6	−1.81083	−0.905414	−	0.424529i	$-0.860440\pi$
$631$	−28702.6	−1.81083	−0.905414	+	0.424529i	$0.860440\pi$
$632$	0	0
$633$	−216.213	−0.0135761
$634$	0	0
$635$	−148.979	−0.00931035
$636$	0	0
$637$	−12650.4	−0.786855
$638$	0	0
$639$	4809.23	0.297731
$640$	0	0
$641$	25013.7	1.54131	0.770655	−	0.637252i	$-0.219929\pi$
$641$	25013.7	1.54131	0.770655	+	0.637252i	$0.219929\pi$
$642$	0	0
$643$	5289.89	0.324437	0.162218	−	0.986755i	$-0.448135\pi$
$643$	5289.89	0.324437	0.162218	+	0.986755i	$0.448135\pi$
$644$	0	0
$645$	403.999	0.0246627
$646$	0	0
$647$	15303.3	0.929883	0.464941	−	0.885341i	$-0.346075\pi$
$647$	15303.3	0.929883	0.464941	+	0.885341i	$0.346075\pi$
$648$	0	0
$649$	801.298	0.0484649
$650$	0	0
$651$	−15.3100	−0.000921728 0
$652$	0	0
$653$	3322.82	0.199130	0.0995651	−	0.995031i	$-0.468255\pi$
$653$	3322.82	0.199130	0.0995651	+	0.995031i	$0.468255\pi$
$654$	0	0
$655$	4720.30	0.281584
$656$	0	0
$657$	1274.82	0.0757006
$658$	0	0
$659$	5863.46	0.346598	0.173299	−	0.984869i	$-0.444557\pi$
$659$	5863.46	0.346598	0.173299	+	0.984869i	$0.444557\pi$
$660$	0	0
$661$	−20323.7	−1.19592	−0.597959	−	0.801527i	$-0.704022\pi$
$661$	−20323.7	−1.19592	−0.597959	+	0.801527i	$0.704022\pi$
$662$	0	0
$663$	426.740	0.0249973
$664$	0	0
$665$	1205.98	0.0703249
$666$	0	0
$667$	249.823	0.0145025
$668$	0	0
$669$	−31.4525	−0.00181767
$670$	0	0
$671$	820.419	0.0472011
$672$	0	0
$673$	5322.74	0.304869	0.152434	−	0.988314i	$-0.451289\pi$
$673$	5322.74	0.304869	0.152434	+	0.988314i	$0.451289\pi$
$674$	0	0
$675$	−216.339	−0.0123361
$676$	0	0
$677$	−8011.51	−0.454811	−0.227406	−	0.973800i	$-0.573024\pi$
$677$	−8011.51	−0.454811	−0.227406	+	0.973800i	$0.573024\pi$
$678$	0	0
$679$	3733.85	0.211034
$680$	0	0
$681$	855.686	0.0481497
$682$	0	0
$683$	2129.90	0.119324	0.0596620	−	0.998219i	$-0.480998\pi$
$683$	2129.90	0.119324	0.0596620	+	0.998219i	$0.480998\pi$
$684$	0	0
$685$	7729.81	0.431155
$686$	0	0
$687$	−80.0557	−0.00444588
$688$	0	0
$689$	−2225.41	−0.123050
$690$	0	0
$691$	−1841.52	−0.101382	−0.0506909	−	0.998714i	$-0.516142\pi$
$691$	−1841.52	−0.101382	−0.0506909	+	0.998714i	$0.516142\pi$
$692$	0	0
$693$	−537.358	−0.0294553
$694$	0	0
$695$	2320.98	0.126676
$696$	0	0
$697$	131.385	0.00713997
$698$	0	0
$699$	−736.140	−0.0398332
$700$	0	0
$701$	542.452	0.0292270	0.0146135	−	0.999893i	$-0.495348\pi$
$701$	542.452	0.0292270	0.0146135	+	0.999893i	$0.495348\pi$
$702$	0	0
$703$	−15005.8	−0.805055
$704$	0	0
$705$	175.829	0.00939308
$706$	0	0
$707$	2737.70	0.145632
$708$	0	0
$709$	6345.43	0.336118	0.168059	−	0.985777i	$-0.446250\pi$
$709$	6345.43	0.336118	0.168059	+	0.985777i	$0.446250\pi$
$710$	0	0
$711$	−29192.3	−1.53980
$712$	0	0
$713$	−970.676	−0.0509847
$714$	0	0
$715$	1648.18	0.0862074
$716$	0	0
$717$	−128.288	−0.00668204
$718$	0	0
$719$	−13487.5	−0.699581	−0.349790	−	0.936828i	$-0.613747\pi$
$719$	−13487.5	−0.699581	−0.349790	+	0.936828i	$0.613747\pi$
$720$	0	0
$721$	2747.63	0.141924
$722$	0	0
$723$	−691.218	−0.0355556
$724$	0	0
$725$	−271.546	−0.0139103
$726$	0	0
$727$	−16549.3	−0.844262	−0.422131	−	0.906535i	$-0.638718\pi$
$727$	−16549.3	−0.844262	−0.422131	+	0.906535i	$0.638718\pi$
$728$	0	0
$729$	−19570.7	−0.994292
$730$	0	0
$731$	−35827.7	−1.81277
$732$	0	0
$733$	15704.0	0.791325	0.395662	−	0.918396i	$-0.370515\pi$
$733$	15704.0	0.791325	0.395662	+	0.918396i	$0.370515\pi$
$734$	0	0
$735$	270.857	0.0135928
$736$	0	0
$737$	8278.34	0.413754
$738$	0	0
$739$	−11247.7	−0.559882	−0.279941	−	0.960017i	$-0.590315\pi$
$739$	−11247.7	−0.559882	−0.279941	+	0.960017i	$0.590315\pi$
$740$	0	0
$741$	639.881	0.0317228
$742$	0	0
$743$	−6515.65	−0.321718	−0.160859	−	0.986977i	$-0.551426\pi$
$743$	−6515.65	−0.321718	−0.160859	+	0.986977i	$0.551426\pi$
$744$	0	0
$745$	12006.7	0.590458
$746$	0	0
$747$	−13070.6	−0.640198
$748$	0	0
$749$	−1993.75	−0.0972630
$750$	0	0
$751$	16811.6	0.816864	0.408432	−	0.912789i	$-0.366076\pi$
$751$	16811.6	0.816864	0.408432	+	0.912789i	$0.366076\pi$
$752$	0	0
$753$	−287.260	−0.0139022
$754$	0	0
$755$	−13345.9	−0.643322
$756$	0	0
$757$	25841.9	1.24074	0.620369	−	0.784310i	$-0.286983\pi$
$757$	25841.9	1.24074	0.620369	+	0.784310i	$0.286983\pi$
$758$	0	0
$759$	32.4660	0.00155262
$760$	0	0
$761$	25579.6	1.21847	0.609237	−	0.792988i	$-0.291476\pi$
$761$	25579.6	1.21847	0.609237	+	0.792988i	$0.291476\pi$
$762$	0	0
$763$	3169.75	0.150397
$764$	0	0
$765$	9588.18	0.453152
$766$	0	0
$767$	−3407.55	−0.160416
$768$	0	0
$769$	−34832.1	−1.63339	−0.816695	−	0.577069i	$-0.804196\pi$
$769$	−34832.1	−1.63339	−0.816695	+	0.577069i	$0.804196\pi$
$770$	0	0
$771$	−1061.48	−0.0495829
$772$	0	0
$773$	34757.8	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$773$	34757.8	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$774$	0	0
$775$	1055.08	0.0489028
$776$	0	0
$777$	51.0664	0.00235778
$778$	0	0
$779$	197.007	0.00906097
$780$	0	0
$781$	1569.71	0.0719187
$782$	0	0
$783$	93.9936	0.00428998
$784$	0	0
$785$	−9733.95	−0.442573
$786$	0	0
$787$	−37002.0	−1.67596	−0.837979	−	0.545702i	$-0.816263\pi$
$787$	−37002.0	−1.67596	−0.837979	+	0.545702i	$0.816263\pi$
$788$	0	0
$789$	1316.12	0.0593856
$790$	0	0
$791$	−5212.24	−0.234293
$792$	0	0
$793$	−3488.86	−0.156233
$794$	0	0
$795$	47.6483	0.00212567
$796$	0	0
$797$	3833.43	0.170373	0.0851863	−	0.996365i	$-0.472851\pi$
$797$	3833.43	0.170373	0.0851863	+	0.996365i	$0.472851\pi$
$798$	0	0
$799$	−15593.0	−0.690415
$800$	0	0
$801$	10581.5	0.466764
$802$	0	0
$803$	416.093	0.0182859
$804$	0	0
$805$	−260.207	−0.0113926
$806$	0	0
$807$	−316.942	−0.0138251
$808$	0	0
$809$	−27596.4	−1.19930	−0.599652	−	0.800261i	$-0.704694\pi$
$809$	−27596.4	−1.19930	−0.599652	+	0.800261i	$0.704694\pi$
$810$	0	0
$811$	−40797.3	−1.76645	−0.883223	−	0.468953i	$-0.844631\pi$
$811$	−40797.3	−1.76645	−0.883223	+	0.468953i	$0.844631\pi$
$812$	0	0
$813$	707.549	0.0305226
$814$	0	0
$815$	−11682.3	−0.502102
$816$	0	0
$817$	−53722.3	−2.30050
$818$	0	0
$819$	2285.13	0.0974957
$820$	0	0
$821$	33441.4	1.42158	0.710788	−	0.703407i	$-0.248339\pi$
$821$	33441.4	1.42158	0.710788	+	0.703407i	$0.248339\pi$
$822$	0	0
$823$	20576.1	0.871493	0.435747	−	0.900069i	$-0.356484\pi$
$823$	20576.1	0.871493	0.435747	+	0.900069i	$0.356484\pi$
$824$	0	0
$825$	−35.2891	−0.00148922
$826$	0	0
$827$	−33197.6	−1.39588	−0.697940	−	0.716156i	$-0.745900\pi$
$827$	−33197.6	−1.39588	−0.697940	+	0.716156i	$0.745900\pi$
$828$	0	0
$829$	22359.4	0.936760	0.468380	−	0.883527i	$-0.344838\pi$
$829$	22359.4	0.936760	0.468380	+	0.883527i	$0.344838\pi$
$830$	0	0
$831$	−730.858	−0.0305092
$832$	0	0
$833$	−24020.3	−0.999105
$834$	0	0
$835$	8937.91	0.370430
$836$	0	0
$837$	−365.208	−0.0150818
$838$	0	0
$839$	−16868.7	−0.694128	−0.347064	−	0.937841i	$-0.612821\pi$
$839$	−16868.7	−0.694128	−0.347064	+	0.937841i	$0.612821\pi$
$840$	0	0
$841$	−24271.0	−0.995163
$842$	0	0
$843$	−516.532	−0.0211035
$844$	0	0
$845$	3976.08	0.161871
$846$	0	0
$847$	2836.22	0.115057
$848$	0	0
$849$	−659.243	−0.0266492
$850$	0	0
$851$	3237.69	0.130419
$852$	0	0
$853$	5686.61	0.228260	0.114130	−	0.993466i	$-0.463592\pi$
$853$	5686.61	0.228260	0.114130	+	0.993466i	$0.463592\pi$
$854$	0	0
$855$	14377.1	0.575072
$856$	0	0
$857$	14995.0	0.597691	0.298845	−	0.954302i	$-0.403399\pi$
$857$	14995.0	0.597691	0.298845	+	0.954302i	$0.403399\pi$
$858$	0	0
$859$	−16013.5	−0.636059	−0.318030	−	0.948081i	$-0.603021\pi$
$859$	−16013.5	−0.636059	−0.318030	+	0.948081i	$0.603021\pi$
$860$	0	0
$861$	−0.670436	−2.65370e−5 0
$862$	0	0
$863$	−9988.20	−0.393977	−0.196989	−	0.980406i	$-0.563116\pi$
$863$	−9988.20	−0.393977	−0.196989	+	0.980406i	$0.563116\pi$
$864$	0	0
$865$	10168.4	0.399695
$866$	0	0
$867$	22.5996	0.000885261 0
$868$	0	0
$869$	−9528.20	−0.371947
$870$	0	0
$871$	−35203.9	−1.36951
$872$	0	0
$873$	44513.0	1.72570
$874$	0	0
$875$	282.833	0.0109274
$876$	0	0
$877$	−24810.0	−0.955271	−0.477636	−	0.878558i	$-0.658506\pi$
$877$	−24810.0	−0.955271	−0.477636	+	0.878558i	$0.658506\pi$
$878$	0	0
$879$	−249.876	−0.00958831
$880$	0	0
$881$	397.093	0.0151855	0.00759274	−	0.999971i	$-0.497583\pi$
$881$	397.093	0.0151855	0.00759274	+	0.999971i	$0.497583\pi$
$882$	0	0
$883$	−10747.7	−0.409613	−0.204807	−	0.978802i	$-0.565657\pi$
$883$	−10747.7	−0.409613	−0.204807	+	0.978802i	$0.565657\pi$
$884$	0	0
$885$	72.9589	0.00277117
$886$	0	0
$887$	51977.0	1.96755	0.983776	−	0.179403i	$-0.0574168\pi$
$887$	51977.0	1.96755	0.983776	+	0.179403i	$0.0574168\pi$
$888$	0	0
$889$	−67.4181	−0.00254346
$890$	0	0
$891$	−6399.98	−0.240637
$892$	0	0
$893$	−23381.1	−0.876170
$894$	0	0
$895$	−19300.8	−0.720842
$896$	0	0
$897$	−138.063	−0.00513910
$898$	0	0
$899$	−458.406	−0.0170063
$900$	0	0
$901$	−4225.57	−0.156242
$902$	0	0
$903$	182.823	0.00673751
$904$	0	0
$905$	13679.5	0.502456
$906$	0	0
$907$	17101.8	0.626080	0.313040	−	0.949740i	$-0.398653\pi$
$907$	17101.8	0.626080	0.313040	+	0.949740i	$0.398653\pi$
$908$	0	0
$909$	32637.4	1.19089
$910$	0	0
$911$	−39489.1	−1.43615	−0.718075	−	0.695966i	$-0.754976\pi$
$911$	−39489.1	−1.43615	−0.718075	+	0.695966i	$0.754976\pi$
$912$	0	0
$913$	−4266.17	−0.154644
$914$	0	0
$915$	74.6999	0.00269891
$916$	0	0
$917$	2136.09	0.0769247
$918$	0	0
$919$	25801.8	0.926140	0.463070	−	0.886322i	$-0.346748\pi$
$919$	25801.8	0.926140	0.463070	+	0.886322i	$0.346748\pi$
$920$	0	0
$921$	−311.805	−0.0111556
$922$	0	0
$923$	−6675.23	−0.238047
$924$	0	0
$925$	−3519.23	−0.125094
$926$	0	0
$927$	32755.8	1.16056
$928$	0	0
$929$	−42120.6	−1.48755	−0.743775	−	0.668430i	$-0.766967\pi$
$929$	−42120.6	−1.48755	−0.743775	+	0.668430i	$0.766967\pi$
$930$	0	0
$931$	−36017.6	−1.26791
$932$	0	0
$933$	218.002	0.00764960
$934$	0	0
$935$	3129.53	0.109462
$936$	0	0
$937$	34261.8	1.19454	0.597270	−	0.802040i	$-0.296252\pi$
$937$	34261.8	1.19454	0.597270	+	0.802040i	$0.296252\pi$
$938$	0	0
$939$	−117.066	−0.00406849
$940$	0	0
$941$	−26424.6	−0.915426	−0.457713	−	0.889100i	$-0.651331\pi$
$941$	−26424.6	−0.915426	−0.457713	+	0.889100i	$0.651331\pi$
$942$	0	0
$943$	−42.5067	−0.00146788
$944$	0	0
$945$	−97.9004	−0.00337005
$946$	0	0
$947$	−4770.21	−0.163686	−0.0818432	−	0.996645i	$-0.526081\pi$
$947$	−4770.21	−0.163686	−0.0818432	+	0.996645i	$0.526081\pi$
$948$	0	0
$949$	−1769.45	−0.0605256
$950$	0	0
$951$	368.581	0.0125679
$952$	0	0
$953$	5571.94	0.189394	0.0946972	−	0.995506i	$-0.469812\pi$
$953$	5571.94	0.189394	0.0946972	+	0.995506i	$0.469812\pi$
$954$	0	0
$955$	14092.6	0.477515
$956$	0	0
$957$	15.3322	0.000517889 0
$958$	0	0
$959$	3498.00	0.117785
$960$	0	0
$961$	−28009.9	−0.940213
$962$	0	0
$963$	−23768.4	−0.795354
$964$	0	0
$965$	−7465.36	−0.249035
$966$	0	0
$967$	−27920.0	−0.928488	−0.464244	−	0.885707i	$-0.653674\pi$
$967$	−27920.0	−0.928488	−0.464244	+	0.885707i	$0.653674\pi$
$968$	0	0
$969$	1214.99	0.0402799
$970$	0	0
$971$	55720.9	1.84157	0.920787	−	0.390065i	$-0.127547\pi$
$971$	55720.9	1.84157	0.920787	+	0.390065i	$0.127547\pi$
$972$	0	0
$973$	1050.32	0.0346061
$974$	0	0
$975$	150.068	0.00492925
$976$	0	0
$977$	−17645.1	−0.577806	−0.288903	−	0.957358i	$-0.593290\pi$
$977$	−17645.1	−0.577806	−0.288903	+	0.957358i	$0.593290\pi$
$978$	0	0
$979$	3453.74	0.112750
$980$	0	0
$981$	37788.0	1.22985
$982$	0	0
$983$	35318.6	1.14597	0.572985	−	0.819566i	$-0.305785\pi$
$983$	35318.6	1.14597	0.572985	+	0.819566i	$0.305785\pi$
$984$	0	0
$985$	−13742.6	−0.444545
$986$	0	0
$987$	79.5686	0.00256606
$988$	0	0
$989$	11591.3	0.372681
$990$	0	0
$991$	−50463.6	−1.61759	−0.808793	−	0.588093i	$-0.799879\pi$
$991$	−50463.6	−1.61759	−0.808793	+	0.588093i	$0.799879\pi$
$992$	0	0
$993$	648.354	0.0207199
$994$	0	0
$995$	17004.4	0.541783
$996$	0	0
$997$	−12815.8	−0.407103	−0.203552	−	0.979064i	$-0.565248\pi$
$997$	−12815.8	−0.407103	−0.203552	+	0.979064i	$0.565248\pi$
$998$	0	0
$999$	1218.15	0.0385792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.b.1.3	✓	6
4.3	odd	2		1840.4.a.s.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.b.1.3	✓	6	1.1	even	1	trivial
1840.4.a.s.1.4		6	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+0.160327 q^{3} -5.00000 q^{5} -2.26267 q^{7} -26.9743 q^{9} -8.80426 q^{11} +37.4404 q^{13} -0.801636 q^{15} +71.0912 q^{17} +106.599 q^{19} -0.362767 q^{21} -23.0000 q^{23} +25.0000 q^{25} -8.65355 q^{27} -10.8619 q^{29} +42.2033 q^{31} -1.41156 q^{33} +11.3133 q^{35} -140.769 q^{37} +6.00272 q^{39} +1.84812 q^{41} -503.969 q^{43} +134.871 q^{45} -219.338 q^{47} -337.880 q^{49} +11.3979 q^{51} -59.4388 q^{53} +44.0213 q^{55} +17.0906 q^{57} -91.0125 q^{59} -93.1843 q^{61} +61.0338 q^{63} -187.202 q^{65} -940.265 q^{67} -3.68753 q^{69} -178.289 q^{71} -47.2604 q^{73} +4.00818 q^{75} +19.9211 q^{77} +1082.23 q^{79} +726.919 q^{81} +484.557 q^{83} -355.456 q^{85} -1.74145 q^{87} -392.280 q^{89} -84.7151 q^{91} +6.76634 q^{93} -532.993 q^{95} -1650.20 q^{97} +237.489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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