LMFDB - Embedded newform 2320.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,4,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2320.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:	$136.884431213$
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} - x^{5} - 91x^{4} + 90x^{3} + 1784x^{2} - 1456x - 3720 $ x^6 - x^5 - 91x^4 + 90x^3 + 1784x^2 - 1456x - 3720
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.14528$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.69185 q^{3} -5.00000 q^{5} +16.0448 q^{7} +32.1646 q^{9} +31.0846 q^{11} +15.0566 q^{13} +38.4593 q^{15} -83.1924 q^{17} -11.6142 q^{19} -123.414 q^{21} -22.9471 q^{23} +25.0000 q^{25} -39.7251 q^{27} -29.0000 q^{29} +117.340 q^{31} -239.098 q^{33} -80.2241 q^{35} -126.486 q^{37} -115.813 q^{39} -500.344 q^{41} +5.84778 q^{43} -160.823 q^{45} +301.232 q^{47} -85.5639 q^{49} +639.903 q^{51} +31.2007 q^{53} -155.423 q^{55} +89.3350 q^{57} -31.3542 q^{59} +80.8436 q^{61} +516.074 q^{63} -75.2828 q^{65} +191.752 q^{67} +176.506 q^{69} -269.462 q^{71} +600.914 q^{73} -192.296 q^{75} +498.746 q^{77} -389.072 q^{79} -562.884 q^{81} +271.037 q^{83} +415.962 q^{85} +223.064 q^{87} -165.976 q^{89} +241.580 q^{91} -902.559 q^{93} +58.0712 q^{95} +1209.58 q^{97} +999.822 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 30 q^{5} - 9 q^{7} + 45 q^{9} + 10 q^{11} + 13 q^{13} - 25 q^{15} - 23 q^{17} - 10 q^{19} - 52 q^{21} - 75 q^{23} + 150 q^{25} + 116 q^{27} - 174 q^{29} + 127 q^{31} - 72 q^{33} + 45 q^{35}+ \cdots + 2976 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 - 30 * q^5 - 9 * q^7 + 45 * q^9 + 10 * q^11 + 13 * q^13 - 25 * q^15 - 23 * q^17 - 10 * q^19 - 52 * q^21 - 75 * q^23 + 150 * q^25 + 116 * q^27 - 174 * q^29 + 127 * q^31 - 72 * q^33 + 45 * q^35 - 158 * q^37 + 407 * q^39 - 622 * q^41 + 311 * q^43 - 225 * q^45 + 628 * q^47 - 879 * q^49 + 886 * q^51 - 813 * q^53 - 50 * q^55 - 1118 * q^57 + 1615 * q^59 - 655 * q^61 + 764 * q^63 - 65 * q^65 + 1000 * q^67 - 1885 * q^69 + 760 * q^71 - 471 * q^73 + 125 * q^75 - 1078 * q^77 + 577 * q^79 - 1402 * q^81 + 1446 * q^83 + 115 * q^85 - 145 * q^87 - 1970 * q^89 + 876 * q^91 - 2274 * q^93 + 50 * q^95 + 1683 * q^97 + 2976 * 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
$3$	−7.69185	−1.48030	−0.740149	−	0.672443i	$-0.765245\pi$
$3$	−7.69185	−1.48030	−0.740149	+	0.672443i	$0.765245\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	16.0448	0.866339	0.433169	−	0.901313i	$-0.357395\pi$
$7$	16.0448	0.866339	0.433169	+	0.901313i	$0.357395\pi$
$8$	0	0
$9$	32.1646	1.19128
$10$	0	0
$11$	31.0846	0.852032	0.426016	−	0.904716i	$-0.359917\pi$
$11$	31.0846	0.852032	0.426016	+	0.904716i	$0.359917\pi$
$12$	0	0
$13$	15.0566	0.321226	0.160613	−	0.987017i	$-0.448653\pi$
$13$	15.0566	0.321226	0.160613	+	0.987017i	$0.448653\pi$
$14$	0	0
$15$	38.4593	0.662009
$16$	0	0
$17$	−83.1924	−1.18689	−0.593444	−	0.804875i	$-0.702232\pi$
$17$	−83.1924	−1.18689	−0.593444	+	0.804875i	$0.702232\pi$
$18$	0	0
$19$	−11.6142	−0.140236	−0.0701181	−	0.997539i	$-0.522338\pi$
$19$	−11.6142	−0.140236	−0.0701181	+	0.997539i	$0.522338\pi$
$20$	0	0
$21$	−123.414	−1.28244
$22$	0	0
$23$	−22.9471	−0.208035	−0.104018	−	0.994575i	$-0.533170\pi$
$23$	−22.9471	−0.208035	−0.104018	+	0.994575i	$0.533170\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−39.7251	−0.283152
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	117.340	0.679833	0.339917	−	0.940456i	$-0.389601\pi$
$31$	117.340	0.679833	0.339917	+	0.940456i	$0.389601\pi$
$32$	0	0
$33$	−239.098	−1.26126
$34$	0	0
$35$	−80.2241	−0.387438
$36$	0	0
$37$	−126.486	−0.562003	−0.281001	−	0.959707i	$-0.590667\pi$
$37$	−126.486	−0.562003	−0.281001	+	0.959707i	$0.590667\pi$
$38$	0	0
$39$	−115.813	−0.475510
$40$	0	0
$41$	−500.344	−1.90587	−0.952935	−	0.303175i	$-0.901953\pi$
$41$	−500.344	−1.90587	−0.952935	+	0.303175i	$0.901953\pi$
$42$	0	0
$43$	5.84778	0.0207390	0.0103695	−	0.999946i	$-0.496699\pi$
$43$	5.84778	0.0207390	0.0103695	+	0.999946i	$0.496699\pi$
$44$	0	0
$45$	−160.823	−0.532757
$46$	0	0
$47$	301.232	0.934876	0.467438	−	0.884026i	$-0.345177\pi$
$47$	301.232	0.934876	0.467438	+	0.884026i	$0.345177\pi$
$48$	0	0
$49$	−85.5639	−0.249458
$50$	0	0
$51$	639.903	1.75695
$52$	0	0
$53$	31.2007	0.0808632	0.0404316	−	0.999182i	$-0.487127\pi$
$53$	31.2007	0.0808632	0.0404316	+	0.999182i	$0.487127\pi$
$54$	0	0
$55$	−155.423	−0.381040
$56$	0	0
$57$	89.3350	0.207591
$58$	0	0
$59$	−31.3542	−0.0691859	−0.0345930	−	0.999401i	$-0.511013\pi$
$59$	−31.3542	−0.0691859	−0.0345930	+	0.999401i	$0.511013\pi$
$60$	0	0
$61$	80.8436	0.169688	0.0848440	−	0.996394i	$-0.472961\pi$
$61$	80.8436	0.169688	0.0848440	+	0.996394i	$0.472961\pi$
$62$	0	0
$63$	516.074	1.03205
$64$	0	0
$65$	−75.2828	−0.143657
$66$	0	0
$67$	191.752	0.349645	0.174822	−	0.984600i	$-0.444065\pi$
$67$	191.752	0.349645	0.174822	+	0.984600i	$0.444065\pi$
$68$	0	0
$69$	176.506	0.307954
$70$	0	0
$71$	−269.462	−0.450412	−0.225206	−	0.974311i	$-0.572306\pi$
$71$	−269.462	−0.450412	−0.225206	+	0.974311i	$0.572306\pi$
$72$	0	0
$73$	600.914	0.963447	0.481724	−	0.876323i	$-0.340011\pi$
$73$	600.914	0.963447	0.481724	+	0.876323i	$0.340011\pi$
$74$	0	0
$75$	−192.296	−0.296059
$76$	0	0
$77$	498.746	0.738148
$78$	0	0
$79$	−389.072	−0.554101	−0.277050	−	0.960855i	$-0.589357\pi$
$79$	−389.072	−0.554101	−0.277050	+	0.960855i	$0.589357\pi$
$80$	0	0
$81$	−562.884	−0.772132
$82$	0	0
$83$	271.037	0.358435	0.179218	−	0.983809i	$-0.442643\pi$
$83$	271.037	0.358435	0.179218	+	0.983809i	$0.442643\pi$
$84$	0	0
$85$	415.962	0.530793
$86$	0	0
$87$	223.064	0.274884
$88$	0	0
$89$	−165.976	−0.197679	−0.0988397	−	0.995103i	$-0.531513\pi$
$89$	−165.976	−0.197679	−0.0988397	+	0.995103i	$0.531513\pi$
$90$	0	0
$91$	241.580	0.278290
$92$	0	0
$93$	−902.559	−1.00636
$94$	0	0
$95$	58.0712	0.0627156
$96$	0	0
$97$	1209.58	1.26612	0.633061	−	0.774102i	$-0.281798\pi$
$97$	1209.58	1.26612	0.633061	+	0.774102i	$0.281798\pi$
$98$	0	0
$99$	999.822	1.01501
$100$	0	0
$101$	1545.49	1.52259	0.761295	−	0.648405i	$-0.224564\pi$
$101$	1545.49	1.52259	0.761295	+	0.648405i	$0.224564\pi$
$102$	0	0
$103$	378.011	0.361617	0.180808	−	0.983518i	$-0.442129\pi$
$103$	378.011	0.361617	0.180808	+	0.983518i	$0.442129\pi$
$104$	0	0
$105$	617.072	0.573524
$106$	0	0
$107$	262.102	0.236807	0.118403	−	0.992966i	$-0.462222\pi$
$107$	262.102	0.236807	0.118403	+	0.992966i	$0.462222\pi$
$108$	0	0
$109$	1337.54	1.17535	0.587676	−	0.809096i	$-0.300043\pi$
$109$	1337.54	1.17535	0.587676	+	0.809096i	$0.300043\pi$
$110$	0	0
$111$	972.908	0.831931
$112$	0	0
$113$	−501.976	−0.417893	−0.208947	−	0.977927i	$-0.567004\pi$
$113$	−501.976	−0.417893	−0.208947	+	0.977927i	$0.567004\pi$
$114$	0	0
$115$	114.736	0.0930362
$116$	0	0
$117$	484.288	0.382670
$118$	0	0
$119$	−1334.81	−1.02825
$120$	0	0
$121$	−364.750	−0.274042
$122$	0	0
$123$	3848.57	2.82125
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1572.60	1.09878	0.549392	−	0.835565i	$-0.314859\pi$
$127$	1572.60	1.09878	0.549392	+	0.835565i	$0.314859\pi$
$128$	0	0
$129$	−44.9802	−0.0306999
$130$	0	0
$131$	487.458	0.325110	0.162555	−	0.986700i	$-0.448027\pi$
$131$	487.458	0.325110	0.162555	+	0.986700i	$0.448027\pi$
$132$	0	0
$133$	−186.348	−0.121492
$134$	0	0
$135$	198.625	0.126629
$136$	0	0
$137$	561.363	0.350076	0.175038	−	0.984562i	$-0.443995\pi$
$137$	561.363	0.350076	0.175038	+	0.984562i	$0.443995\pi$
$138$	0	0
$139$	−527.373	−0.321807	−0.160904	−	0.986970i	$-0.551441\pi$
$139$	−527.373	−0.321807	−0.160904	+	0.986970i	$0.551441\pi$
$140$	0	0
$141$	−2317.03	−1.38389
$142$	0	0
$143$	468.027	0.273695
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	658.145	0.369271
$148$	0	0
$149$	190.524	0.104754	0.0523768	−	0.998627i	$-0.483320\pi$
$149$	190.524	0.104754	0.0523768	+	0.998627i	$0.483320\pi$
$150$	0	0
$151$	1056.81	0.569550	0.284775	−	0.958594i	$-0.408081\pi$
$151$	1056.81	0.569550	0.284775	+	0.958594i	$0.408081\pi$
$152$	0	0
$153$	−2675.85	−1.41392
$154$	0	0
$155$	−586.698	−0.304031
$156$	0	0
$157$	−708.890	−0.360354	−0.180177	−	0.983634i	$-0.557667\pi$
$157$	−708.890	−0.360354	−0.180177	+	0.983634i	$0.557667\pi$
$158$	0	0
$159$	−239.991	−0.119702
$160$	0	0
$161$	−368.183	−0.180229
$162$	0	0
$163$	443.505	0.213116	0.106558	−	0.994306i	$-0.466017\pi$
$163$	443.505	0.213116	0.106558	+	0.994306i	$0.466017\pi$
$164$	0	0
$165$	1195.49	0.564053
$166$	0	0
$167$	−3330.52	−1.54325	−0.771627	−	0.636075i	$-0.780557\pi$
$167$	−3330.52	−1.54325	−0.771627	+	0.636075i	$0.780557\pi$
$168$	0	0
$169$	−1970.30	−0.896814
$170$	0	0
$171$	−373.567	−0.167061
$172$	0	0
$173$	−2160.18	−0.949339	−0.474670	−	0.880164i	$-0.657432\pi$
$173$	−2160.18	−0.949339	−0.474670	+	0.880164i	$0.657432\pi$
$174$	0	0
$175$	401.120	0.173268
$176$	0	0
$177$	241.172	0.102416
$178$	0	0
$179$	108.508	0.0453087	0.0226544	−	0.999743i	$-0.492788\pi$
$179$	108.508	0.0453087	0.0226544	+	0.999743i	$0.492788\pi$
$180$	0	0
$181$	687.869	0.282480	0.141240	−	0.989975i	$-0.454891\pi$
$181$	687.869	0.282480	0.141240	+	0.989975i	$0.454891\pi$
$182$	0	0
$183$	−621.837	−0.251189
$184$	0	0
$185$	632.428	0.251335
$186$	0	0
$187$	−2586.00	−1.01127
$188$	0	0
$189$	−637.381	−0.245305
$190$	0	0
$191$	−3943.02	−1.49375	−0.746877	−	0.664963i	$-0.768447\pi$
$191$	−3943.02	−1.49375	−0.746877	+	0.664963i	$0.768447\pi$
$192$	0	0
$193$	−3124.06	−1.16516	−0.582578	−	0.812775i	$-0.697956\pi$
$193$	−3124.06	−1.16516	−0.582578	+	0.812775i	$0.697956\pi$
$194$	0	0
$195$	579.064	0.212655
$196$	0	0
$197$	−2176.53	−0.787164	−0.393582	−	0.919290i	$-0.628764\pi$
$197$	−2176.53	−0.787164	−0.393582	+	0.919290i	$0.628764\pi$
$198$	0	0
$199$	1250.92	0.445604	0.222802	−	0.974864i	$-0.428480\pi$
$199$	1250.92	0.445604	0.222802	+	0.974864i	$0.428480\pi$
$200$	0	0
$201$	−1474.93	−0.517578
$202$	0	0
$203$	−465.300	−0.160875
$204$	0	0
$205$	2501.72	0.852331
$206$	0	0
$207$	−738.085	−0.247828
$208$	0	0
$209$	−361.024	−0.119486
$210$	0	0
$211$	−3838.31	−1.25232	−0.626162	−	0.779693i	$-0.715375\pi$
$211$	−3838.31	−1.25232	−0.626162	+	0.779693i	$0.715375\pi$
$212$	0	0
$213$	2072.66	0.666744
$214$	0	0
$215$	−29.2389	−0.00927477
$216$	0	0
$217$	1882.69	0.588966
$218$	0	0
$219$	−4622.14	−1.42619
$220$	0	0
$221$	−1252.59	−0.381260
$222$	0	0
$223$	−1723.40	−0.517521	−0.258761	−	0.965942i	$-0.583314\pi$
$223$	−1723.40	−0.517521	−0.258761	+	0.965942i	$0.583314\pi$
$224$	0	0
$225$	804.114	0.238256
$226$	0	0
$227$	3406.40	0.995995	0.497997	−	0.867179i	$-0.334069\pi$
$227$	3406.40	0.995995	0.497997	+	0.867179i	$0.334069\pi$
$228$	0	0
$229$	4538.85	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$229$	4538.85	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$230$	0	0
$231$	−3836.28	−1.09268
$232$	0	0
$233$	414.870	0.116648	0.0583242	−	0.998298i	$-0.481424\pi$
$233$	414.870	0.116648	0.0583242	+	0.998298i	$0.481424\pi$
$234$	0	0
$235$	−1506.16	−0.418089
$236$	0	0
$237$	2992.68	0.820234
$238$	0	0
$239$	1307.91	0.353982	0.176991	−	0.984212i	$-0.443364\pi$
$239$	1307.91	0.353982	0.176991	+	0.984212i	$0.443364\pi$
$240$	0	0
$241$	−3942.50	−1.05377	−0.526886	−	0.849936i	$-0.676641\pi$
$241$	−3942.50	−1.05377	−0.526886	+	0.849936i	$0.676641\pi$
$242$	0	0
$243$	5402.20	1.42614
$244$	0	0
$245$	427.820	0.111561
$246$	0	0
$247$	−174.870	−0.0450475
$248$	0	0
$249$	−2084.77	−0.530591
$250$	0	0
$251$	−921.189	−0.231653	−0.115827	−	0.993269i	$-0.536952\pi$
$251$	−921.189	−0.231653	−0.115827	+	0.993269i	$0.536952\pi$
$252$	0	0
$253$	−713.302	−0.177253
$254$	0	0
$255$	−3199.52	−0.785731
$256$	0	0
$257$	−679.151	−0.164841	−0.0824207	−	0.996598i	$-0.526265\pi$
$257$	−679.151	−0.164841	−0.0824207	+	0.996598i	$0.526265\pi$
$258$	0	0
$259$	−2029.44	−0.486885
$260$	0	0
$261$	−932.772	−0.221215
$262$	0	0
$263$	−5846.60	−1.37079	−0.685393	−	0.728173i	$-0.740370\pi$
$263$	−5846.60	−1.37079	−0.685393	+	0.728173i	$0.740370\pi$
$264$	0	0
$265$	−156.004	−0.0361631
$266$	0	0
$267$	1276.67	0.292624
$268$	0	0
$269$	−921.923	−0.208961	−0.104481	−	0.994527i	$-0.533318\pi$
$269$	−921.923	−0.208961	−0.104481	+	0.994527i	$0.533318\pi$
$270$	0	0
$271$	−6533.88	−1.46459	−0.732297	−	0.680985i	$-0.761552\pi$
$271$	−6533.88	−1.46459	−0.732297	+	0.680985i	$0.761552\pi$
$272$	0	0
$273$	−1858.19	−0.411953
$274$	0	0
$275$	777.114	0.170406
$276$	0	0
$277$	6758.46	1.46598	0.732990	−	0.680239i	$-0.238124\pi$
$277$	6758.46	1.46598	0.732990	+	0.680239i	$0.238124\pi$
$278$	0	0
$279$	3774.18	0.809872
$280$	0	0
$281$	−9049.56	−1.92118	−0.960590	−	0.277970i	$-0.910338\pi$
$281$	−9049.56	−1.92118	−0.960590	+	0.277970i	$0.910338\pi$
$282$	0	0
$283$	−7660.93	−1.60917	−0.804585	−	0.593837i	$-0.797612\pi$
$283$	−7660.93	−1.60917	−0.804585	+	0.593837i	$0.797612\pi$
$284$	0	0
$285$	−446.675	−0.0928377
$286$	0	0
$287$	−8027.93	−1.65113
$288$	0	0
$289$	2007.97	0.408705
$290$	0	0
$291$	−9303.88	−1.87424
$292$	0	0
$293$	4977.90	0.992533	0.496267	−	0.868170i	$-0.334704\pi$
$293$	4977.90	0.992533	0.496267	+	0.868170i	$0.334704\pi$
$294$	0	0
$295$	156.771	0.0309409
$296$	0	0
$297$	−1234.84	−0.241254
$298$	0	0
$299$	−345.505	−0.0668263
$300$	0	0
$301$	93.8265	0.0179670
$302$	0	0
$303$	−11887.7	−2.25389
$304$	0	0
$305$	−404.218	−0.0758868
$306$	0	0
$307$	171.110	0.0318103	0.0159052	−	0.999874i	$-0.494937\pi$
$307$	171.110	0.0318103	0.0159052	+	0.999874i	$0.494937\pi$
$308$	0	0
$309$	−2907.60	−0.535300
$310$	0	0
$311$	3881.41	0.707700	0.353850	−	0.935302i	$-0.384872\pi$
$311$	3881.41	0.707700	0.353850	+	0.935302i	$0.384872\pi$
$312$	0	0
$313$	3578.45	0.646217	0.323108	−	0.946362i	$-0.395272\pi$
$313$	3578.45	0.646217	0.323108	+	0.946362i	$0.395272\pi$
$314$	0	0
$315$	−2580.37	−0.461548
$316$	0	0
$317$	4025.99	0.713319	0.356660	−	0.934234i	$-0.383916\pi$
$317$	4025.99	0.713319	0.356660	+	0.934234i	$0.383916\pi$
$318$	0	0
$319$	−901.452	−0.158218
$320$	0	0
$321$	−2016.05	−0.350544
$322$	0	0
$323$	966.216	0.166445
$324$	0	0
$325$	376.414	0.0642452
$326$	0	0
$327$	−10288.2	−1.73987
$328$	0	0
$329$	4833.21	0.809919
$330$	0	0
$331$	446.811	0.0741962	0.0370981	−	0.999312i	$-0.488189\pi$
$331$	446.811	0.0741962	0.0370981	+	0.999312i	$0.488189\pi$
$332$	0	0
$333$	−4068.35	−0.669503
$334$	0	0
$335$	−958.758	−0.156366
$336$	0	0
$337$	3235.03	0.522918	0.261459	−	0.965215i	$-0.415796\pi$
$337$	3235.03	0.522918	0.261459	+	0.965215i	$0.415796\pi$
$338$	0	0
$339$	3861.13	0.618607
$340$	0	0
$341$	3647.45	0.579240
$342$	0	0
$343$	−6876.23	−1.08245
$344$	0	0
$345$	−882.530	−0.137721
$346$	0	0
$347$	−3281.37	−0.507646	−0.253823	−	0.967251i	$-0.581688\pi$
$347$	−3281.37	−0.507646	−0.253823	+	0.967251i	$0.581688\pi$
$348$	0	0
$349$	−8576.39	−1.31543	−0.657713	−	0.753269i	$-0.728476\pi$
$349$	−8576.39	−1.31543	−0.657713	+	0.753269i	$0.728476\pi$
$350$	0	0
$351$	−598.123	−0.0909556
$352$	0	0
$353$	−12116.6	−1.82692	−0.913459	−	0.406931i	$-0.866599\pi$
$353$	−12116.6	−1.82692	−0.913459	+	0.406931i	$0.866599\pi$
$354$	0	0
$355$	1347.31	0.201431
$356$	0	0
$357$	10267.1	1.52211
$358$	0	0
$359$	8234.49	1.21058	0.605292	−	0.796004i	$-0.293056\pi$
$359$	8234.49	1.21058	0.605292	+	0.796004i	$0.293056\pi$
$360$	0	0
$361$	−6724.11	−0.980334
$362$	0	0
$363$	2805.60	0.405663
$364$	0	0
$365$	−3004.57	−0.430867
$366$	0	0
$367$	−6135.07	−0.872611	−0.436306	−	0.899799i	$-0.643713\pi$
$367$	−6135.07	−0.872611	−0.436306	+	0.899799i	$0.643713\pi$
$368$	0	0
$369$	−16093.4	−2.27042
$370$	0	0
$371$	500.610	0.0700549
$372$	0	0
$373$	5421.05	0.752524	0.376262	−	0.926513i	$-0.377209\pi$
$373$	5421.05	0.752524	0.376262	+	0.926513i	$0.377209\pi$
$374$	0	0
$375$	961.481	0.132402
$376$	0	0
$377$	−436.640	−0.0596502
$378$	0	0
$379$	13190.8	1.78777	0.893884	−	0.448298i	$-0.147969\pi$
$379$	13190.8	1.78777	0.893884	+	0.448298i	$0.147969\pi$
$380$	0	0
$381$	−12096.2	−1.62653
$382$	0	0
$383$	8801.65	1.17426	0.587132	−	0.809491i	$-0.300257\pi$
$383$	8801.65	1.17426	0.587132	+	0.809491i	$0.300257\pi$
$384$	0	0
$385$	−2493.73	−0.330110
$386$	0	0
$387$	188.091	0.0247060
$388$	0	0
$389$	−7098.28	−0.925185	−0.462593	−	0.886571i	$-0.653081\pi$
$389$	−7098.28	−0.925185	−0.462593	+	0.886571i	$0.653081\pi$
$390$	0	0
$391$	1909.03	0.246915
$392$	0	0
$393$	−3749.45	−0.481259
$394$	0	0
$395$	1945.36	0.247801
$396$	0	0
$397$	−983.121	−0.124286	−0.0621428	−	0.998067i	$-0.519793\pi$
$397$	−983.121	−0.124286	−0.0621428	+	0.998067i	$0.519793\pi$
$398$	0	0
$399$	1433.36	0.179844
$400$	0	0
$401$	−11566.4	−1.44039	−0.720196	−	0.693771i	$-0.755948\pi$
$401$	−11566.4	−1.44039	−0.720196	+	0.693771i	$0.755948\pi$
$402$	0	0
$403$	1766.73	0.218380
$404$	0	0
$405$	2814.42	0.345308
$406$	0	0
$407$	−3931.75	−0.478844
$408$	0	0
$409$	−5076.95	−0.613787	−0.306893	−	0.951744i	$-0.599290\pi$
$409$	−5076.95	−0.613787	−0.306893	+	0.951744i	$0.599290\pi$
$410$	0	0
$411$	−4317.92	−0.518217
$412$	0	0
$413$	−503.072	−0.0599384
$414$	0	0
$415$	−1355.18	−0.160297
$416$	0	0
$417$	4056.48	0.476370
$418$	0	0
$419$	7054.26	0.822490	0.411245	−	0.911525i	$-0.365094\pi$
$419$	7054.26	0.822490	0.411245	+	0.911525i	$0.365094\pi$
$420$	0	0
$421$	−47.7238	−0.00552474	−0.00276237	−	0.999996i	$-0.500879\pi$
$421$	−47.7238	−0.00552474	−0.00276237	+	0.999996i	$0.500879\pi$
$422$	0	0
$423$	9688.99	1.11370
$424$	0	0
$425$	−2079.81	−0.237378
$426$	0	0
$427$	1297.12	0.147007
$428$	0	0
$429$	−3599.99	−0.405150
$430$	0	0
$431$	−11596.3	−1.29600	−0.647999	−	0.761641i	$-0.724394\pi$
$431$	−11596.3	−1.29600	−0.647999	+	0.761641i	$0.724394\pi$
$432$	0	0
$433$	3017.20	0.334867	0.167434	−	0.985883i	$-0.446452\pi$
$433$	3017.20	0.334867	0.167434	+	0.985883i	$0.446452\pi$
$434$	0	0
$435$	−1115.32	−0.122932
$436$	0	0
$437$	266.514	0.0291741
$438$	0	0
$439$	2030.54	0.220757	0.110379	−	0.993890i	$-0.464794\pi$
$439$	2030.54	0.220757	0.110379	+	0.993890i	$0.464794\pi$
$440$	0	0
$441$	−2752.13	−0.297174
$442$	0	0
$443$	2622.89	0.281303	0.140652	−	0.990059i	$-0.455080\pi$
$443$	2622.89	0.281303	0.140652	+	0.990059i	$0.455080\pi$
$444$	0	0
$445$	829.882	0.0884049
$446$	0	0
$447$	−1465.48	−0.155067
$448$	0	0
$449$	−8289.68	−0.871301	−0.435651	−	0.900116i	$-0.643482\pi$
$449$	−8289.68	−0.871301	−0.435651	+	0.900116i	$0.643482\pi$
$450$	0	0
$451$	−15553.0	−1.62386
$452$	0	0
$453$	−8128.83	−0.843103
$454$	0	0
$455$	−1207.90	−0.124455
$456$	0	0
$457$	11291.3	1.15577	0.577885	−	0.816118i	$-0.303878\pi$
$457$	11291.3	1.15577	0.577885	+	0.816118i	$0.303878\pi$
$458$	0	0
$459$	3304.82	0.336069
$460$	0	0
$461$	−4292.63	−0.433683	−0.216841	−	0.976207i	$-0.569575\pi$
$461$	−4292.63	−0.433683	−0.216841	+	0.976207i	$0.569575\pi$
$462$	0	0
$463$	−4006.35	−0.402140	−0.201070	−	0.979577i	$-0.564442\pi$
$463$	−4006.35	−0.402140	−0.201070	+	0.979577i	$0.564442\pi$
$464$	0	0
$465$	4512.80	0.450056
$466$	0	0
$467$	−7220.10	−0.715432	−0.357716	−	0.933831i	$-0.616444\pi$
$467$	−7220.10	−0.715432	−0.357716	+	0.933831i	$0.616444\pi$
$468$	0	0
$469$	3076.62	0.302911
$470$	0	0
$471$	5452.67	0.533431
$472$	0	0
$473$	181.776	0.0176703
$474$	0	0
$475$	−290.356	−0.0280472
$476$	0	0
$477$	1003.56	0.0963307
$478$	0	0
$479$	16656.9	1.58888	0.794438	−	0.607346i	$-0.207766\pi$
$479$	16656.9	1.58888	0.794438	+	0.607346i	$0.207766\pi$
$480$	0	0
$481$	−1904.44	−0.180530
$482$	0	0
$483$	2832.01	0.266792
$484$	0	0
$485$	−6047.88	−0.566227
$486$	0	0
$487$	10289.5	0.957412	0.478706	−	0.877975i	$-0.341106\pi$
$487$	10289.5	0.957412	0.478706	+	0.877975i	$0.341106\pi$
$488$	0	0
$489$	−3411.37	−0.315475
$490$	0	0
$491$	−14263.3	−1.31099	−0.655493	−	0.755201i	$-0.727539\pi$
$491$	−14263.3	−1.31099	−0.655493	+	0.755201i	$0.727539\pi$
$492$	0	0
$493$	2412.58	0.220400
$494$	0	0
$495$	−4999.11	−0.453926
$496$	0	0
$497$	−4323.47	−0.390210
$498$	0	0
$499$	−1843.00	−0.165339	−0.0826693	−	0.996577i	$-0.526345\pi$
$499$	−1843.00	−0.165339	−0.0826693	+	0.996577i	$0.526345\pi$
$500$	0	0
$501$	25617.9	2.28448
$502$	0	0
$503$	4350.87	0.385678	0.192839	−	0.981230i	$-0.438231\pi$
$503$	4350.87	0.385678	0.192839	+	0.981230i	$0.438231\pi$
$504$	0	0
$505$	−7727.43	−0.680923
$506$	0	0
$507$	15155.3	1.32755
$508$	0	0
$509$	16783.1	1.46148	0.730742	−	0.682653i	$-0.239174\pi$
$509$	16783.1	1.46148	0.730742	+	0.682653i	$0.239174\pi$
$510$	0	0
$511$	9641.55	0.834671
$512$	0	0
$513$	461.376	0.0397081
$514$	0	0
$515$	−1890.06	−0.161720
$516$	0	0
$517$	9363.66	0.796544
$518$	0	0
$519$	16615.8	1.40530
$520$	0	0
$521$	−4014.47	−0.337576	−0.168788	−	0.985652i	$-0.553985\pi$
$521$	−4014.47	−0.337576	−0.168788	+	0.985652i	$0.553985\pi$
$522$	0	0
$523$	−15944.4	−1.33308	−0.666538	−	0.745471i	$-0.732225\pi$
$523$	−15944.4	−1.33308	−0.666538	+	0.745471i	$0.732225\pi$
$524$	0	0
$525$	−3085.36	−0.256488
$526$	0	0
$527$	−9761.76	−0.806886
$528$	0	0
$529$	−11640.4	−0.956721
$530$	0	0
$531$	−1008.49	−0.0824198
$532$	0	0
$533$	−7533.46	−0.612215
$534$	0	0
$535$	−1310.51	−0.105903
$536$	0	0
$537$	−834.627	−0.0670704
$538$	0	0
$539$	−2659.72	−0.212546
$540$	0	0
$541$	−20434.2	−1.62391	−0.811954	−	0.583721i	$-0.801596\pi$
$541$	−20434.2	−1.62391	−0.811954	+	0.583721i	$0.801596\pi$
$542$	0	0
$543$	−5290.99	−0.418155
$544$	0	0
$545$	−6687.72	−0.525634
$546$	0	0
$547$	−6626.20	−0.517945	−0.258973	−	0.965885i	$-0.583384\pi$
$547$	−6626.20	−0.517945	−0.258973	+	0.965885i	$0.583384\pi$
$548$	0	0
$549$	2600.30	0.202146
$550$	0	0
$551$	336.813	0.0260412
$552$	0	0
$553$	−6242.58	−0.480039
$554$	0	0
$555$	−4864.54	−0.372051
$556$	0	0
$557$	−10434.1	−0.793727	−0.396864	−	0.917878i	$-0.629901\pi$
$557$	−10434.1	−0.793727	−0.396864	+	0.917878i	$0.629901\pi$
$558$	0	0
$559$	88.0474	0.00666191
$560$	0	0
$561$	19891.1	1.49698
$562$	0	0
$563$	7846.64	0.587383	0.293691	−	0.955900i	$-0.405116\pi$
$563$	7846.64	0.587383	0.293691	+	0.955900i	$0.405116\pi$
$564$	0	0
$565$	2509.88	0.186888
$566$	0	0
$567$	−9031.37	−0.668927
$568$	0	0
$569$	−25122.3	−1.85093	−0.925465	−	0.378832i	$-0.876326\pi$
$569$	−25122.3	−1.85093	−0.925465	+	0.378832i	$0.876326\pi$
$570$	0	0
$571$	−11500.3	−0.842862	−0.421431	−	0.906861i	$-0.638472\pi$
$571$	−11500.3	−0.842862	−0.421431	+	0.906861i	$0.638472\pi$
$572$	0	0
$573$	30329.1	2.21120
$574$	0	0
$575$	−573.679	−0.0416071
$576$	0	0
$577$	−9164.70	−0.661233	−0.330617	−	0.943765i	$-0.607257\pi$
$577$	−9164.70	−0.661233	−0.330617	+	0.943765i	$0.607257\pi$
$578$	0	0
$579$	24029.8	1.72478
$580$	0	0
$581$	4348.73	0.310526
$582$	0	0
$583$	969.861	0.0688980
$584$	0	0
$585$	−2421.44	−0.171135
$586$	0	0
$587$	4624.15	0.325143	0.162572	−	0.986697i	$-0.448021\pi$
$587$	4624.15	0.325143	0.162572	+	0.986697i	$0.448021\pi$
$588$	0	0
$589$	−1362.81	−0.0953373
$590$	0	0
$591$	16741.5	1.16524
$592$	0	0
$593$	−17603.7	−1.21905	−0.609526	−	0.792766i	$-0.708640\pi$
$593$	−17603.7	−1.21905	−0.609526	+	0.792766i	$0.708640\pi$
$594$	0	0
$595$	6674.03	0.459846
$596$	0	0
$597$	−9621.86	−0.659626
$598$	0	0
$599$	−16865.8	−1.15044	−0.575222	−	0.817997i	$-0.695084\pi$
$599$	−16865.8	−1.15044	−0.575222	+	0.817997i	$0.695084\pi$
$600$	0	0
$601$	13746.3	0.932987	0.466494	−	0.884525i	$-0.345517\pi$
$601$	13746.3	0.932987	0.466494	+	0.884525i	$0.345517\pi$
$602$	0	0
$603$	6167.61	0.416525
$604$	0	0
$605$	1823.75	0.122555
$606$	0	0
$607$	9415.05	0.629564	0.314782	−	0.949164i	$-0.398069\pi$
$607$	9415.05	0.629564	0.314782	+	0.949164i	$0.398069\pi$
$608$	0	0
$609$	3579.02	0.238143
$610$	0	0
$611$	4535.52	0.300307
$612$	0	0
$613$	−11753.3	−0.774409	−0.387205	−	0.921994i	$-0.626559\pi$
$613$	−11753.3	−0.774409	−0.387205	+	0.921994i	$0.626559\pi$
$614$	0	0
$615$	−19242.9	−1.26170
$616$	0	0
$617$	11464.2	0.748027	0.374014	−	0.927423i	$-0.377981\pi$
$617$	11464.2	0.748027	0.374014	+	0.927423i	$0.377981\pi$
$618$	0	0
$619$	7416.72	0.481588	0.240794	−	0.970576i	$-0.422592\pi$
$619$	7416.72	0.481588	0.240794	+	0.970576i	$0.422592\pi$
$620$	0	0
$621$	911.577	0.0589055
$622$	0	0
$623$	−2663.06	−0.171257
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2776.94	0.176874
$628$	0	0
$629$	10522.6	0.667035
$630$	0	0
$631$	13552.5	0.855020	0.427510	−	0.904011i	$-0.359391\pi$
$631$	13552.5	0.855020	0.427510	+	0.904011i	$0.359391\pi$
$632$	0	0
$633$	29523.7	1.85381
$634$	0	0
$635$	−7862.99	−0.491391
$636$	0	0
$637$	−1288.30	−0.0801322
$638$	0	0
$639$	−8667.13	−0.536567
$640$	0	0
$641$	−21456.3	−1.32211	−0.661055	−	0.750337i	$-0.729891\pi$
$641$	−21456.3	−1.32211	−0.661055	+	0.750337i	$0.729891\pi$
$642$	0	0
$643$	5794.95	0.355413	0.177706	−	0.984084i	$-0.443132\pi$
$643$	5794.95	0.355413	0.177706	+	0.984084i	$0.443132\pi$
$644$	0	0
$645$	224.901	0.0137294
$646$	0	0
$647$	13848.5	0.841486	0.420743	−	0.907180i	$-0.361769\pi$
$647$	13848.5	0.841486	0.420743	+	0.907180i	$0.361769\pi$
$648$	0	0
$649$	−974.632	−0.0589486
$650$	0	0
$651$	−14481.4	−0.871844
$652$	0	0
$653$	−7835.40	−0.469561	−0.234780	−	0.972048i	$-0.575437\pi$
$653$	−7835.40	−0.469561	−0.234780	+	0.972048i	$0.575437\pi$
$654$	0	0
$655$	−2437.29	−0.145394
$656$	0	0
$657$	19328.1	1.14774
$658$	0	0
$659$	−21053.2	−1.24449	−0.622243	−	0.782824i	$-0.713779\pi$
$659$	−21053.2	−1.24449	−0.622243	+	0.782824i	$0.713779\pi$
$660$	0	0
$661$	2587.65	0.152266	0.0761330	−	0.997098i	$-0.475743\pi$
$661$	2587.65	0.152266	0.0761330	+	0.997098i	$0.475743\pi$
$662$	0	0
$663$	9634.74	0.564377
$664$	0	0
$665$	931.741	0.0543329
$666$	0	0
$667$	665.467	0.0386312
$668$	0	0
$669$	13256.1	0.766085
$670$	0	0
$671$	2512.99	0.144580
$672$	0	0
$673$	20359.3	1.16611	0.583057	−	0.812431i	$-0.301856\pi$
$673$	20359.3	1.16611	0.583057	+	0.812431i	$0.301856\pi$
$674$	0	0
$675$	−993.127	−0.0566303
$676$	0	0
$677$	16020.3	0.909471	0.454736	−	0.890627i	$-0.349734\pi$
$677$	16020.3	0.909471	0.454736	+	0.890627i	$0.349734\pi$
$678$	0	0
$679$	19407.4	1.09689
$680$	0	0
$681$	−26201.5	−1.47437
$682$	0	0
$683$	11079.7	0.620722	0.310361	−	0.950619i	$-0.399550\pi$
$683$	11079.7	0.620722	0.310361	+	0.950619i	$0.399550\pi$
$684$	0	0
$685$	−2806.81	−0.156559
$686$	0	0
$687$	−34912.2	−1.93884
$688$	0	0
$689$	469.775	0.0259753
$690$	0	0
$691$	1573.50	0.0866265	0.0433133	−	0.999062i	$-0.486209\pi$
$691$	1573.50	0.0866265	0.0433133	+	0.999062i	$0.486209\pi$
$692$	0	0
$693$	16042.0	0.879341
$694$	0	0
$695$	2636.87	0.143917
$696$	0	0
$697$	41624.8	2.26206
$698$	0	0
$699$	−3191.12	−0.172674
$700$	0	0
$701$	−28271.6	−1.52326	−0.761630	−	0.648012i	$-0.775600\pi$
$701$	−28271.6	−1.52326	−0.761630	+	0.648012i	$0.775600\pi$
$702$	0	0
$703$	1469.03	0.0788132
$704$	0	0
$705$	11585.2	0.618897
$706$	0	0
$707$	24797.0	1.31908
$708$	0	0
$709$	−14638.5	−0.775401	−0.387700	−	0.921785i	$-0.626730\pi$
$709$	−14638.5	−0.775401	−0.387700	+	0.921785i	$0.626730\pi$
$710$	0	0
$711$	−12514.3	−0.660089
$712$	0	0
$713$	−2692.61	−0.141429
$714$	0	0
$715$	−2340.13	−0.122400
$716$	0	0
$717$	−10060.3	−0.523999
$718$	0	0
$719$	2609.41	0.135347	0.0676735	−	0.997708i	$-0.478442\pi$
$719$	2609.41	0.135347	0.0676735	+	0.997708i	$0.478442\pi$
$720$	0	0
$721$	6065.12	0.313283
$722$	0	0
$723$	30325.2	1.55990
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−8386.64	−0.427845	−0.213923	−	0.976851i	$-0.568624\pi$
$727$	−8386.64	−0.427845	−0.213923	+	0.976851i	$0.568624\pi$
$728$	0	0
$729$	−26355.0	−1.33897
$730$	0	0
$731$	−486.490	−0.0246149
$732$	0	0
$733$	29969.0	1.51014	0.755068	−	0.655646i	$-0.227604\pi$
$733$	29969.0	1.51014	0.755068	+	0.655646i	$0.227604\pi$
$734$	0	0
$735$	−3290.73	−0.165143
$736$	0	0
$737$	5960.52	0.297908
$738$	0	0
$739$	−908.616	−0.0452286	−0.0226143	−	0.999744i	$-0.507199\pi$
$739$	−908.616	−0.0452286	−0.0226143	+	0.999744i	$0.507199\pi$
$740$	0	0
$741$	1345.08	0.0666837
$742$	0	0
$743$	32141.4	1.58702	0.793508	−	0.608560i	$-0.208253\pi$
$743$	32141.4	1.58702	0.793508	+	0.608560i	$0.208253\pi$
$744$	0	0
$745$	−952.618	−0.0468473
$746$	0	0
$747$	8717.77	0.426997
$748$	0	0
$749$	4205.37	0.205155
$750$	0	0
$751$	−9075.10	−0.440952	−0.220476	−	0.975392i	$-0.570761\pi$
$751$	−9075.10	−0.440952	−0.220476	+	0.975392i	$0.570761\pi$
$752$	0	0
$753$	7085.65	0.342915
$754$	0	0
$755$	−5284.05	−0.254711
$756$	0	0
$757$	−28009.3	−1.34480	−0.672402	−	0.740187i	$-0.734737\pi$
$757$	−28009.3	−1.34480	−0.672402	+	0.740187i	$0.734737\pi$
$758$	0	0
$759$	5486.61	0.262387
$760$	0	0
$761$	−18706.3	−0.891066	−0.445533	−	0.895265i	$-0.646986\pi$
$761$	−18706.3	−0.891066	−0.445533	+	0.895265i	$0.646986\pi$
$762$	0	0
$763$	21460.6	1.01825
$764$	0	0
$765$	13379.2	0.632323
$766$	0	0
$767$	−472.086	−0.0222243
$768$	0	0
$769$	7774.27	0.364561	0.182280	−	0.983247i	$-0.441652\pi$
$769$	7774.27	0.364561	0.182280	+	0.983247i	$0.441652\pi$
$770$	0	0
$771$	5223.92	0.244014
$772$	0	0
$773$	−24000.1	−1.11672	−0.558359	−	0.829600i	$-0.688569\pi$
$773$	−24000.1	−1.11672	−0.558359	+	0.829600i	$0.688569\pi$
$774$	0	0
$775$	2933.49	0.135967
$776$	0	0
$777$	15610.1	0.720734
$778$	0	0
$779$	5811.12	0.267272
$780$	0	0
$781$	−8376.12	−0.383766
$782$	0	0
$783$	1152.03	0.0525799
$784$	0	0
$785$	3544.45	0.161155
$786$	0	0
$787$	−20749.3	−0.939814	−0.469907	−	0.882716i	$-0.655713\pi$
$787$	−20749.3	−0.939814	−0.469907	+	0.882716i	$0.655713\pi$
$788$	0	0
$789$	44971.2	2.02917
$790$	0	0
$791$	−8054.12	−0.362037
$792$	0	0
$793$	1217.23	0.0545082
$794$	0	0
$795$	1199.96	0.0535321
$796$	0	0
$797$	1546.40	0.0687282	0.0343641	−	0.999409i	$-0.489059\pi$
$797$	1546.40	0.0687282	0.0343641	+	0.999409i	$0.489059\pi$
$798$	0	0
$799$	−25060.2	−1.10959
$800$	0	0
$801$	−5338.56	−0.235491
$802$	0	0
$803$	18679.1	0.820888
$804$	0	0
$805$	1840.91	0.0806008
$806$	0	0
$807$	7091.29	0.309325
$808$	0	0
$809$	14261.2	0.619773	0.309886	−	0.950774i	$-0.399709\pi$
$809$	14261.2	0.619773	0.309886	+	0.950774i	$0.399709\pi$
$810$	0	0
$811$	8098.05	0.350630	0.175315	−	0.984512i	$-0.443906\pi$
$811$	8098.05	0.350630	0.175315	+	0.984512i	$0.443906\pi$
$812$	0	0
$813$	50257.7	2.16804
$814$	0	0
$815$	−2217.52	−0.0953085
$816$	0	0
$817$	−67.9175	−0.00290836
$818$	0	0
$819$	7770.31	0.331522
$820$	0	0
$821$	−31734.9	−1.34903	−0.674517	−	0.738259i	$-0.735648\pi$
$821$	−31734.9	−1.34903	−0.674517	+	0.738259i	$0.735648\pi$
$822$	0	0
$823$	−9300.47	−0.393918	−0.196959	−	0.980412i	$-0.563107\pi$
$823$	−9300.47	−0.393918	−0.196959	+	0.980412i	$0.563107\pi$
$824$	0	0
$825$	−5977.45	−0.252252
$826$	0	0
$827$	19533.5	0.821338	0.410669	−	0.911784i	$-0.365295\pi$
$827$	19533.5	0.821338	0.410669	+	0.911784i	$0.365295\pi$
$828$	0	0
$829$	1108.39	0.0464366	0.0232183	−	0.999730i	$-0.492609\pi$
$829$	1108.39	0.0464366	0.0232183	+	0.999730i	$0.492609\pi$
$830$	0	0
$831$	−51985.1	−2.17009
$832$	0	0
$833$	7118.27	0.296078
$834$	0	0
$835$	16652.6	0.690164
$836$	0	0
$837$	−4661.33	−0.192496
$838$	0	0
$839$	−40285.1	−1.65768	−0.828841	−	0.559484i	$-0.810999\pi$
$839$	−40285.1	−1.65768	−0.828841	+	0.559484i	$0.810999\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	69607.9	2.84392
$844$	0	0
$845$	9851.50	0.401067
$846$	0	0
$847$	−5852.34	−0.237413
$848$	0	0
$849$	58926.8	2.38205
$850$	0	0
$851$	2902.48	0.116916
$852$	0	0
$853$	−4959.80	−0.199086	−0.0995430	−	0.995033i	$-0.531738\pi$
$853$	−4959.80	−0.199086	−0.0995430	+	0.995033i	$0.531738\pi$
$854$	0	0
$855$	1867.83	0.0747118
$856$	0	0
$857$	2961.63	0.118048	0.0590241	−	0.998257i	$-0.481201\pi$
$857$	2961.63	0.118048	0.0590241	+	0.998257i	$0.481201\pi$
$858$	0	0
$859$	10657.4	0.423312	0.211656	−	0.977344i	$-0.432114\pi$
$859$	10657.4	0.423312	0.211656	+	0.977344i	$0.432114\pi$
$860$	0	0
$861$	61749.7	2.44416
$862$	0	0
$863$	−10244.1	−0.404072	−0.202036	−	0.979378i	$-0.564756\pi$
$863$	−10244.1	−0.404072	−0.202036	+	0.979378i	$0.564756\pi$
$864$	0	0
$865$	10800.9	0.424557
$866$	0	0
$867$	−15445.0	−0.605005
$868$	0	0
$869$	−12094.1	−0.472112
$870$	0	0
$871$	2887.12	0.112315
$872$	0	0
$873$	38905.5	1.50831
$874$	0	0
$875$	−2005.60	−0.0774877
$876$	0	0
$877$	−43671.7	−1.68151	−0.840757	−	0.541413i	$-0.817889\pi$
$877$	−43671.7	−1.68151	−0.840757	+	0.541413i	$0.817889\pi$
$878$	0	0
$879$	−38289.3	−1.46924
$880$	0	0
$881$	−37652.8	−1.43990	−0.719952	−	0.694024i	$-0.755836\pi$
$881$	−37652.8	−1.43990	−0.719952	+	0.694024i	$0.755836\pi$
$882$	0	0
$883$	26609.6	1.01414	0.507070	−	0.861905i	$-0.330728\pi$
$883$	26609.6	1.01414	0.507070	+	0.861905i	$0.330728\pi$
$884$	0	0
$885$	−1205.86	−0.0458017
$886$	0	0
$887$	49149.8	1.86053	0.930265	−	0.366889i	$-0.119577\pi$
$887$	49149.8	1.86053	0.930265	+	0.366889i	$0.119577\pi$
$888$	0	0
$889$	25232.1	0.951919
$890$	0	0
$891$	−17497.0	−0.657881
$892$	0	0
$893$	−3498.58	−0.131104
$894$	0	0
$895$	−542.540	−0.0202627
$896$	0	0
$897$	2657.57	0.0989228
$898$	0	0
$899$	−3402.85	−0.126242
$900$	0	0
$901$	−2595.66	−0.0959756
$902$	0	0
$903$	−721.700	−0.0265965
$904$	0	0
$905$	−3439.35	−0.126329
$906$	0	0
$907$	−36813.3	−1.34770	−0.673852	−	0.738867i	$-0.735361\pi$
$907$	−36813.3	−1.34770	−0.673852	+	0.738867i	$0.735361\pi$
$908$	0	0
$909$	49709.9	1.81383
$910$	0	0
$911$	−17126.7	−0.622868	−0.311434	−	0.950268i	$-0.600809\pi$
$911$	−17126.7	−0.622868	−0.311434	+	0.950268i	$0.600809\pi$
$912$	0	0
$913$	8425.05	0.305398
$914$	0	0
$915$	3109.18	0.112335
$916$	0	0
$917$	7821.17	0.281655
$918$	0	0
$919$	−30450.8	−1.09301	−0.546507	−	0.837455i	$-0.684043\pi$
$919$	−30450.8	−1.09301	−0.546507	+	0.837455i	$0.684043\pi$
$920$	0	0
$921$	−1316.15	−0.0470888
$922$	0	0
$923$	−4057.17	−0.144684
$924$	0	0
$925$	−3162.14	−0.112401
$926$	0	0
$927$	12158.6	0.430787
$928$	0	0
$929$	−19086.8	−0.674077	−0.337038	−	0.941491i	$-0.609425\pi$
$929$	−19086.8	−0.674077	−0.337038	+	0.941491i	$0.609425\pi$
$930$	0	0
$931$	993.760	0.0349830
$932$	0	0
$933$	−29855.3	−1.04761
$934$	0	0
$935$	12930.0	0.452252
$936$	0	0
$937$	17125.4	0.597079	0.298540	−	0.954397i	$-0.403500\pi$
$937$	17125.4	0.597079	0.298540	+	0.954397i	$0.403500\pi$
$938$	0	0
$939$	−27524.9	−0.956593
$940$	0	0
$941$	−34308.1	−1.18854	−0.594269	−	0.804267i	$-0.702558\pi$
$941$	−34308.1	−1.18854	−0.594269	+	0.804267i	$0.702558\pi$
$942$	0	0
$943$	11481.5	0.396488
$944$	0	0
$945$	3186.91	0.109704
$946$	0	0
$947$	4103.52	0.140809	0.0704046	−	0.997519i	$-0.477571\pi$
$947$	4103.52	0.140809	0.0704046	+	0.997519i	$0.477571\pi$
$948$	0	0
$949$	9047.69	0.309484
$950$	0	0
$951$	−30967.3	−1.05592
$952$	0	0
$953$	24975.5	0.848936	0.424468	−	0.905443i	$-0.360461\pi$
$953$	24975.5	0.848936	0.424468	+	0.905443i	$0.360461\pi$
$954$	0	0
$955$	19715.1	0.668027
$956$	0	0
$957$	6933.84	0.234210
$958$	0	0
$959$	9006.96	0.303285
$960$	0	0
$961$	−16022.4	−0.537827
$962$	0	0
$963$	8430.38	0.282103
$964$	0	0
$965$	15620.3	0.521073
$966$	0	0
$967$	−16482.8	−0.548140	−0.274070	−	0.961710i	$-0.588370\pi$
$967$	−16482.8	−0.548140	−0.274070	+	0.961710i	$0.588370\pi$
$968$	0	0
$969$	−7431.99	−0.246388
$970$	0	0
$971$	40016.5	1.32254	0.661272	−	0.750146i	$-0.270017\pi$
$971$	40016.5	1.32254	0.661272	+	0.750146i	$0.270017\pi$
$972$	0	0
$973$	−8461.60	−0.278794
$974$	0	0
$975$	−2895.32	−0.0951020
$976$	0	0
$977$	43518.5	1.42506	0.712529	−	0.701643i	$-0.247550\pi$
$977$	43518.5	1.42506	0.712529	+	0.701643i	$0.247550\pi$
$978$	0	0
$979$	−5159.30	−0.168429
$980$	0	0
$981$	43021.5	1.40017
$982$	0	0
$983$	−31243.2	−1.01374	−0.506869	−	0.862023i	$-0.669197\pi$
$983$	−31243.2	−1.01374	−0.506869	+	0.862023i	$0.669197\pi$
$984$	0	0
$985$	10882.6	0.352030
$986$	0	0
$987$	−37176.3	−1.19892
$988$	0	0
$989$	−134.190	−0.00431445
$990$	0	0
$991$	−20703.7	−0.663648	−0.331824	−	0.943341i	$-0.607664\pi$
$991$	−20703.7	−0.663648	−0.331824	+	0.943341i	$0.607664\pi$
$992$	0	0
$993$	−3436.80	−0.109832
$994$	0	0
$995$	−6254.58	−0.199280
$996$	0	0
$997$	57742.0	1.83421	0.917104	−	0.398647i	$-0.130520\pi$
$997$	57742.0	1.83421	0.917104	+	0.398647i	$0.130520\pi$
$998$	0	0
$999$	5024.65	0.159132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.4.a.o.1.1		6
4.3	odd	2		580.4.a.a.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.4.a.a.1.6	✓	6	4.3	odd	2
2320.4.a.o.1.1		6	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q-7.69185 q^{3} -5.00000 q^{5} +16.0448 q^{7} +32.1646 q^{9} +31.0846 q^{11} +15.0566 q^{13} +38.4593 q^{15} -83.1924 q^{17} -11.6142 q^{19} -123.414 q^{21} -22.9471 q^{23} +25.0000 q^{25} -39.7251 q^{27} -29.0000 q^{29} +117.340 q^{31} -239.098 q^{33} -80.2241 q^{35} -126.486 q^{37} -115.813 q^{39} -500.344 q^{41} +5.84778 q^{43} -160.823 q^{45} +301.232 q^{47} -85.5639 q^{49} +639.903 q^{51} +31.2007 q^{53} -155.423 q^{55} +89.3350 q^{57} -31.3542 q^{59} +80.8436 q^{61} +516.074 q^{63} -75.2828 q^{65} +191.752 q^{67} +176.506 q^{69} -269.462 q^{71} +600.914 q^{73} -192.296 q^{75} +498.746 q^{77} -389.072 q^{79} -562.884 q^{81} +271.037 q^{83} +415.962 q^{85} +223.064 q^{87} -165.976 q^{89} +241.580 q^{91} -902.559 q^{93} +58.0712 q^{95} +1209.58 q^{97} +999.822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 30 q^{5} - 9 q^{7} + 45 q^{9} + 10 q^{11} + 13 q^{13} - 25 q^{15} - 23 q^{17} - 10 q^{19} - 52 q^{21} - 75 q^{23} + 150 q^{25} + 116 q^{27} - 174 q^{29} + 127 q^{31} - 72 q^{33} + 45 q^{35}+ \cdots + 2976 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 - 30 * q^5 - 9 * q^7 + 45 * q^9 + 10 * q^11 + 13 * q^13 - 25 * q^15 - 23 * q^17 - 10 * q^19 - 52 * q^21 - 75 * q^23 + 150 * q^25 + 116 * q^27 - 174 * q^29 + 127 * q^31 - 72 * q^33 + 45 * q^35 - 158 * q^37 + 407 * q^39 - 622 * q^41 + 311 * q^43 - 225 * q^45 + 628 * q^47 - 879 * q^49 + 886 * q^51 - 813 * q^53 - 50 * q^55 - 1118 * q^57 + 1615 * q^59 - 655 * q^61 + 764 * q^63 - 65 * q^65 + 1000 * q^67 - 1885 * q^69 + 760 * q^71 - 471 * q^73 + 125 * q^75 - 1078 * q^77 + 577 * q^79 - 1402 * q^81 + 1446 * q^83 + 115 * q^85 - 145 * q^87 - 1970 * q^89 + 876 * q^91 - 2274 * q^93 + 50 * q^95 + 1683 * q^97 + 2976 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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