LMFDB - Embedded newform 2320.4.a.o.1.6

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			1.6
Root			$-4.88726$ 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)

$f(q)$	$=$	$q+8.48308 q^{3} -5.00000 q^{5} -7.07143 q^{7} +44.9627 q^{9} +49.6170 q^{11} +11.3703 q^{13} -42.4154 q^{15} -43.7733 q^{17} -140.693 q^{19} -59.9876 q^{21} -171.311 q^{23} +25.0000 q^{25} +152.379 q^{27} -29.0000 q^{29} -24.3789 q^{31} +420.905 q^{33} +35.3572 q^{35} +62.2739 q^{37} +96.4552 q^{39} -161.107 q^{41} -350.585 q^{43} -224.813 q^{45} -211.265 q^{47} -292.995 q^{49} -371.332 q^{51} -581.165 q^{53} -248.085 q^{55} -1193.51 q^{57} +609.450 q^{59} -169.936 q^{61} -317.951 q^{63} -56.8515 q^{65} +322.156 q^{67} -1453.25 q^{69} +419.068 q^{71} -138.501 q^{73} +212.077 q^{75} -350.863 q^{77} -15.7298 q^{79} +78.6508 q^{81} +1051.81 q^{83} +218.866 q^{85} -246.009 q^{87} +606.725 q^{89} -80.4044 q^{91} -206.808 q^{93} +703.464 q^{95} +1506.62 q^{97} +2230.91 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	8.48308	1.63257	0.816285	−	0.577649i	$-0.196030\pi$
$3$	8.48308	1.63257	0.816285	+	0.577649i	$0.196030\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.07143	−0.381822	−0.190911	−	0.981607i	$-0.561144\pi$
$7$	−7.07143	−0.381822	−0.190911	+	0.981607i	$0.561144\pi$
$8$	0	0
$9$	44.9627	1.66528
$10$	0	0
$11$	49.6170	1.36001	0.680004	−	0.733209i	$-0.261978\pi$
$11$	49.6170	1.36001	0.680004	+	0.733209i	$0.261978\pi$
$12$	0	0
$13$	11.3703	0.242581	0.121291	−	0.992617i	$-0.461297\pi$
$13$	11.3703	0.242581	0.121291	+	0.992617i	$0.461297\pi$
$14$	0	0
$15$	−42.4154	−0.730107
$16$	0	0
$17$	−43.7733	−0.624505	−0.312252	−	0.949999i	$-0.601083\pi$
$17$	−43.7733	−0.624505	−0.312252	+	0.949999i	$0.601083\pi$
$18$	0	0
$19$	−140.693	−1.69880	−0.849398	−	0.527753i	$-0.823035\pi$
$19$	−140.693	−1.69880	−0.849398	+	0.527753i	$0.823035\pi$
$20$	0	0
$21$	−59.9876	−0.623350
$22$	0	0
$23$	−171.311	−1.55308	−0.776539	−	0.630069i	$-0.783027\pi$
$23$	−171.311	−1.55308	−0.776539	+	0.630069i	$0.783027\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.379	1.08612
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−24.3789	−0.141245	−0.0706223	−	0.997503i	$-0.522499\pi$
$31$	−24.3789	−0.141245	−0.0706223	+	0.997503i	$0.522499\pi$
$32$	0	0
$33$	420.905	2.22031
$34$	0	0
$35$	35.3572	0.170756
$36$	0	0
$37$	62.2739	0.276696	0.138348	−	0.990384i	$-0.455821\pi$
$37$	62.2739	0.276696	0.138348	+	0.990384i	$0.455821\pi$
$38$	0	0
$39$	96.4552	0.396031
$40$	0	0
$41$	−161.107	−0.613675	−0.306838	−	0.951762i	$-0.599271\pi$
$41$	−161.107	−0.613675	−0.306838	+	0.951762i	$0.599271\pi$
$42$	0	0
$43$	−350.585	−1.24334	−0.621671	−	0.783279i	$-0.713546\pi$
$43$	−350.585	−1.24334	−0.621671	+	0.783279i	$0.713546\pi$
$44$	0	0
$45$	−224.813	−0.744738
$46$	0	0
$47$	−211.265	−0.655664	−0.327832	−	0.944736i	$-0.606318\pi$
$47$	−211.265	−0.655664	−0.327832	+	0.944736i	$0.606318\pi$
$48$	0	0
$49$	−292.995	−0.854212
$50$	0	0
$51$	−371.332	−1.01955
$52$	0	0
$53$	−581.165	−1.50621	−0.753106	−	0.657900i	$-0.771445\pi$
$53$	−581.165	−1.50621	−0.753106	+	0.657900i	$0.771445\pi$
$54$	0	0
$55$	−248.085	−0.608214
$56$	0	0
$57$	−1193.51	−2.77340
$58$	0	0
$59$	609.450	1.34481	0.672404	−	0.740184i	$-0.265262\pi$
$59$	609.450	1.34481	0.672404	+	0.740184i	$0.265262\pi$
$60$	0	0
$61$	−169.936	−0.356690	−0.178345	−	0.983968i	$-0.557074\pi$
$61$	−169.936	−0.356690	−0.178345	+	0.983968i	$0.557074\pi$
$62$	0	0
$63$	−317.951	−0.635842
$64$	0	0
$65$	−56.8515	−0.108486
$66$	0	0
$67$	322.156	0.587428	0.293714	−	0.955893i	$-0.405109\pi$
$67$	322.156	0.587428	0.293714	+	0.955893i	$0.405109\pi$
$68$	0	0
$69$	−1453.25	−2.53551
$70$	0	0
$71$	419.068	0.700483	0.350241	−	0.936660i	$-0.386100\pi$
$71$	419.068	0.700483	0.350241	+	0.936660i	$0.386100\pi$
$72$	0	0
$73$	−138.501	−0.222059	−0.111030	−	0.993817i	$-0.535415\pi$
$73$	−138.501	−0.222059	−0.111030	+	0.993817i	$0.535415\pi$
$74$	0	0
$75$	212.077	0.326514
$76$	0	0
$77$	−350.863	−0.519280
$78$	0	0
$79$	−15.7298	−0.0224017	−0.0112009	−	0.999937i	$-0.503565\pi$
$79$	−15.7298	−0.0224017	−0.0112009	+	0.999937i	$0.503565\pi$
$80$	0	0
$81$	78.6508	0.107889
$82$	0	0
$83$	1051.81	1.39097	0.695487	−	0.718538i	$-0.255189\pi$
$83$	1051.81	1.39097	0.695487	+	0.718538i	$0.255189\pi$
$84$	0	0
$85$	218.866	0.279287
$86$	0	0
$87$	−246.009	−0.303161
$88$	0	0
$89$	606.725	0.722615	0.361307	−	0.932447i	$-0.382331\pi$
$89$	606.725	0.722615	0.361307	+	0.932447i	$0.382331\pi$
$90$	0	0
$91$	−80.4044	−0.0926227
$92$	0	0
$93$	−206.808	−0.230592
$94$	0	0
$95$	703.464	0.759725
$96$	0	0
$97$	1506.62	1.57705	0.788525	−	0.615002i	$-0.210845\pi$
$97$	1506.62	1.57705	0.788525	+	0.615002i	$0.210845\pi$
$98$	0	0
$99$	2230.91	2.26480
$100$	0	0
$101$	−788.748	−0.777063	−0.388531	−	0.921436i	$-0.627017\pi$
$101$	−788.748	−0.777063	−0.388531	+	0.921436i	$0.627017\pi$
$102$	0	0
$103$	342.031	0.327198	0.163599	−	0.986527i	$-0.447690\pi$
$103$	342.031	0.327198	0.163599	+	0.986527i	$0.447690\pi$
$104$	0	0
$105$	299.938	0.278771
$106$	0	0
$107$	−29.0515	−0.0262478	−0.0131239	−	0.999914i	$-0.504178\pi$
$107$	−29.0515	−0.0262478	−0.0131239	+	0.999914i	$0.504178\pi$
$108$	0	0
$109$	−1294.70	−1.13771	−0.568853	−	0.822439i	$-0.692613\pi$
$109$	−1294.70	−1.13771	−0.568853	+	0.822439i	$0.692613\pi$
$110$	0	0
$111$	528.275	0.451726
$112$	0	0
$113$	1072.95	0.893228	0.446614	−	0.894727i	$-0.352630\pi$
$113$	1072.95	0.893228	0.446614	+	0.894727i	$0.352630\pi$
$114$	0	0
$115$	856.555	0.694558
$116$	0	0
$117$	511.240	0.403967
$118$	0	0
$119$	309.540	0.238449
$120$	0	0
$121$	1130.84	0.849621
$122$	0	0
$123$	−1366.68	−1.00187
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1534.40	−1.07210	−0.536048	−	0.844188i	$-0.680083\pi$
$127$	−1534.40	−1.07210	−0.536048	+	0.844188i	$0.680083\pi$
$128$	0	0
$129$	−2974.04	−2.02984
$130$	0	0
$131$	−2216.05	−1.47800	−0.738998	−	0.673707i	$-0.764701\pi$
$131$	−2216.05	−1.47800	−0.738998	+	0.673707i	$0.764701\pi$
$132$	0	0
$133$	994.899	0.648637
$134$	0	0
$135$	−761.895	−0.485729
$136$	0	0
$137$	−1791.50	−1.11721	−0.558606	−	0.829433i	$-0.688663\pi$
$137$	−1791.50	−1.11721	−0.558606	+	0.829433i	$0.688663\pi$
$138$	0	0
$139$	1597.56	0.974844	0.487422	−	0.873167i	$-0.337937\pi$
$139$	1597.56	0.974844	0.487422	+	0.873167i	$0.337937\pi$
$140$	0	0
$141$	−1792.18	−1.07042
$142$	0	0
$143$	564.160	0.329912
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−2485.50	−1.39456
$148$	0	0
$149$	−3028.56	−1.66516	−0.832581	−	0.553903i	$-0.813138\pi$
$149$	−3028.56	−1.66516	−0.832581	+	0.553903i	$0.813138\pi$
$150$	0	0
$151$	1004.38	0.541293	0.270646	−	0.962679i	$-0.412763\pi$
$151$	1004.38	0.541293	0.270646	+	0.962679i	$0.412763\pi$
$152$	0	0
$153$	−1968.16	−1.03998
$154$	0	0
$155$	121.895	0.0631665
$156$	0	0
$157$	−1590.05	−0.808281	−0.404141	−	0.914697i	$-0.632429\pi$
$157$	−1590.05	−0.808281	−0.404141	+	0.914697i	$0.632429\pi$
$158$	0	0
$159$	−4930.07	−2.45900
$160$	0	0
$161$	1211.41	0.592999
$162$	0	0
$163$	670.551	0.322219	0.161109	−	0.986937i	$-0.448493\pi$
$163$	670.551	0.322219	0.161109	+	0.986937i	$0.448493\pi$
$164$	0	0
$165$	−2104.52	−0.992952
$166$	0	0
$167$	397.647	0.184257	0.0921283	−	0.995747i	$-0.470633\pi$
$167$	397.647	0.184257	0.0921283	+	0.995747i	$0.470633\pi$
$168$	0	0
$169$	−2067.72	−0.941154
$170$	0	0
$171$	−6325.92	−2.82898
$172$	0	0
$173$	2673.82	1.17507	0.587534	−	0.809199i	$-0.300099\pi$
$173$	2673.82	1.17507	0.587534	+	0.809199i	$0.300099\pi$
$174$	0	0
$175$	−176.786	−0.0763643
$176$	0	0
$177$	5170.02	2.19549
$178$	0	0
$179$	−375.124	−0.156637	−0.0783186	−	0.996928i	$-0.524955\pi$
$179$	−375.124	−0.156637	−0.0783186	+	0.996928i	$0.524955\pi$
$180$	0	0
$181$	−2962.43	−1.21655	−0.608276	−	0.793726i	$-0.708139\pi$
$181$	−2962.43	−1.21655	−0.608276	+	0.793726i	$0.708139\pi$
$182$	0	0
$183$	−1441.58	−0.582322
$184$	0	0
$185$	−311.369	−0.123742
$186$	0	0
$187$	−2171.90	−0.849331
$188$	0	0
$189$	−1077.54	−0.414706
$190$	0	0
$191$	−3178.65	−1.20418	−0.602092	−	0.798427i	$-0.705666\pi$
$191$	−3178.65	−1.20418	−0.602092	+	0.798427i	$0.705666\pi$
$192$	0	0
$193$	271.618	0.101303	0.0506515	−	0.998716i	$-0.483870\pi$
$193$	271.618	0.101303	0.0506515	+	0.998716i	$0.483870\pi$
$194$	0	0
$195$	−482.276	−0.177110
$196$	0	0
$197$	2936.50	1.06202	0.531008	−	0.847367i	$-0.321813\pi$
$197$	2936.50	1.06202	0.531008	+	0.847367i	$0.321813\pi$
$198$	0	0
$199$	−4011.65	−1.42904	−0.714519	−	0.699616i	$-0.753354\pi$
$199$	−4011.65	−1.42904	−0.714519	+	0.699616i	$0.753354\pi$
$200$	0	0
$201$	2732.88	0.959017
$202$	0	0
$203$	205.072	0.0709025
$204$	0	0
$205$	805.535	0.274444
$206$	0	0
$207$	−7702.60	−2.58632
$208$	0	0
$209$	−6980.75	−2.31038
$210$	0	0
$211$	5321.80	1.73634	0.868171	−	0.496266i	$-0.165296\pi$
$211$	5321.80	1.73634	0.868171	+	0.496266i	$0.165296\pi$
$212$	0	0
$213$	3554.99	1.14359
$214$	0	0
$215$	1752.92	0.556039
$216$	0	0
$217$	172.394	0.0539302
$218$	0	0
$219$	−1174.92	−0.362527
$220$	0	0
$221$	−497.716	−0.151493
$222$	0	0
$223$	−4132.12	−1.24084	−0.620420	−	0.784270i	$-0.713038\pi$
$223$	−4132.12	−1.24084	−0.620420	+	0.784270i	$0.713038\pi$
$224$	0	0
$225$	1124.07	0.333057
$226$	0	0
$227$	6160.58	1.80129	0.900643	−	0.434559i	$-0.143096\pi$
$227$	6160.58	1.80129	0.900643	+	0.434559i	$0.143096\pi$
$228$	0	0
$229$	−5702.79	−1.64564	−0.822819	−	0.568304i	$-0.807600\pi$
$229$	−5702.79	−1.64564	−0.822819	+	0.568304i	$0.807600\pi$
$230$	0	0
$231$	−2976.40	−0.847761
$232$	0	0
$233$	−2533.81	−0.712427	−0.356214	−	0.934405i	$-0.615932\pi$
$233$	−2533.81	−0.712427	−0.356214	+	0.934405i	$0.615932\pi$
$234$	0	0
$235$	1056.33	0.293222
$236$	0	0
$237$	−133.437	−0.0365724
$238$	0	0
$239$	−339.594	−0.0919101	−0.0459550	−	0.998944i	$-0.514633\pi$
$239$	−339.594	−0.0919101	−0.0459550	+	0.998944i	$0.514633\pi$
$240$	0	0
$241$	2490.58	0.665693	0.332847	−	0.942981i	$-0.391991\pi$
$241$	2490.58	0.665693	0.332847	+	0.942981i	$0.391991\pi$
$242$	0	0
$243$	−3447.03	−0.909988
$244$	0	0
$245$	1464.97	0.382015
$246$	0	0
$247$	−1599.72	−0.412096
$248$	0	0
$249$	8922.57	2.27086
$250$	0	0
$251$	−1934.93	−0.486581	−0.243291	−	0.969953i	$-0.578227\pi$
$251$	−1934.93	−0.486581	−0.243291	+	0.969953i	$0.578227\pi$
$252$	0	0
$253$	−8499.93	−2.11220
$254$	0	0
$255$	1856.66	0.455956
$256$	0	0
$257$	−741.416	−0.179954	−0.0899772	−	0.995944i	$-0.528679\pi$
$257$	−741.416	−0.179954	−0.0899772	+	0.995944i	$0.528679\pi$
$258$	0	0
$259$	−440.366	−0.105649
$260$	0	0
$261$	−1303.92	−0.309236
$262$	0	0
$263$	4691.78	1.10003	0.550014	−	0.835155i	$-0.314622\pi$
$263$	4691.78	1.10003	0.550014	+	0.835155i	$0.314622\pi$
$264$	0	0
$265$	2905.83	0.673598
$266$	0	0
$267$	5146.90	1.17972
$268$	0	0
$269$	−6449.25	−1.46178	−0.730888	−	0.682498i	$-0.760894\pi$
$269$	−6449.25	−1.46178	−0.730888	+	0.682498i	$0.760894\pi$
$270$	0	0
$271$	5486.65	1.22985	0.614926	−	0.788585i	$-0.289186\pi$
$271$	5486.65	1.22985	0.614926	+	0.788585i	$0.289186\pi$
$272$	0	0
$273$	−682.077	−0.151213
$274$	0	0
$275$	1240.42	0.272002
$276$	0	0
$277$	−4441.54	−0.963417	−0.481708	−	0.876332i	$-0.659984\pi$
$277$	−4441.54	−0.963417	−0.481708	+	0.876332i	$0.659984\pi$
$278$	0	0
$279$	−1096.14	−0.235212
$280$	0	0
$281$	492.309	0.104515	0.0522575	−	0.998634i	$-0.483358\pi$
$281$	492.309	0.104515	0.0522575	+	0.998634i	$0.483358\pi$
$282$	0	0
$283$	−4850.65	−1.01887	−0.509437	−	0.860508i	$-0.670146\pi$
$283$	−4850.65	−1.01887	−0.509437	+	0.860508i	$0.670146\pi$
$284$	0	0
$285$	5967.54	1.24030
$286$	0	0
$287$	1139.26	0.234314
$288$	0	0
$289$	−2996.90	−0.609994
$290$	0	0
$291$	12780.8	2.57465
$292$	0	0
$293$	6560.79	1.30814	0.654071	−	0.756433i	$-0.273060\pi$
$293$	6560.79	1.30814	0.654071	+	0.756433i	$0.273060\pi$
$294$	0	0
$295$	−3047.25	−0.601416
$296$	0	0
$297$	7560.58	1.47714
$298$	0	0
$299$	−1947.86	−0.376748
$300$	0	0
$301$	2479.14	0.474734
$302$	0	0
$303$	−6691.01	−1.26861
$304$	0	0
$305$	849.681	0.159517
$306$	0	0
$307$	−7308.77	−1.35874	−0.679370	−	0.733796i	$-0.737747\pi$
$307$	−7308.77	−1.35874	−0.679370	+	0.733796i	$0.737747\pi$
$308$	0	0
$309$	2901.48	0.534173
$310$	0	0
$311$	761.424	0.138831	0.0694154	−	0.997588i	$-0.477887\pi$
$311$	761.424	0.138831	0.0694154	+	0.997588i	$0.477887\pi$
$312$	0	0
$313$	5238.58	0.946014	0.473007	−	0.881059i	$-0.343169\pi$
$313$	5238.58	0.946014	0.473007	+	0.881059i	$0.343169\pi$
$314$	0	0
$315$	1589.75	0.284357
$316$	0	0
$317$	−8054.16	−1.42702	−0.713512	−	0.700643i	$-0.752897\pi$
$317$	−8054.16	−1.42702	−0.713512	+	0.700643i	$0.752897\pi$
$318$	0	0
$319$	−1438.89	−0.252547
$320$	0	0
$321$	−246.446	−0.0428513
$322$	0	0
$323$	6158.58	1.06091
$324$	0	0
$325$	284.258	0.0485162
$326$	0	0
$327$	−10983.1	−1.85738
$328$	0	0
$329$	1493.95	0.250347
$330$	0	0
$331$	115.772	0.0192249	0.00961243	−	0.999954i	$-0.496940\pi$
$331$	115.772	0.0192249	0.00961243	+	0.999954i	$0.496940\pi$
$332$	0	0
$333$	2800.00	0.460778
$334$	0	0
$335$	−1610.78	−0.262706
$336$	0	0
$337$	5101.77	0.824662	0.412331	−	0.911034i	$-0.364715\pi$
$337$	5101.77	0.824662	0.412331	+	0.911034i	$0.364715\pi$
$338$	0	0
$339$	9101.94	1.45826
$340$	0	0
$341$	−1209.61	−0.192094
$342$	0	0
$343$	4497.40	0.707978
$344$	0	0
$345$	7266.23	1.13391
$346$	0	0
$347$	−7932.92	−1.22727	−0.613633	−	0.789591i	$-0.710293\pi$
$347$	−7932.92	−1.22727	−0.613633	+	0.789591i	$0.710293\pi$
$348$	0	0
$349$	−3840.43	−0.589035	−0.294518	−	0.955646i	$-0.595159\pi$
$349$	−3840.43	−0.589035	−0.294518	+	0.955646i	$0.595159\pi$
$350$	0	0
$351$	1732.60	0.263473
$352$	0	0
$353$	−5282.31	−0.796456	−0.398228	−	0.917287i	$-0.630375\pi$
$353$	−5282.31	−0.796456	−0.398228	+	0.917287i	$0.630375\pi$
$354$	0	0
$355$	−2095.34	−0.313265
$356$	0	0
$357$	2625.85	0.389285
$358$	0	0
$359$	5124.61	0.753389	0.376694	−	0.926338i	$-0.377061\pi$
$359$	5124.61	0.753389	0.376694	+	0.926338i	$0.377061\pi$
$360$	0	0
$361$	12935.4	1.88591
$362$	0	0
$363$	9593.05	1.38706
$364$	0	0
$365$	692.505	0.0993078
$366$	0	0
$367$	5397.49	0.767702	0.383851	−	0.923395i	$-0.374598\pi$
$367$	5397.49	0.767702	0.383851	+	0.923395i	$0.374598\pi$
$368$	0	0
$369$	−7243.81	−1.02194
$370$	0	0
$371$	4109.67	0.575104
$372$	0	0
$373$	−2075.12	−0.288058	−0.144029	−	0.989573i	$-0.546006\pi$
$373$	−2075.12	−0.288058	−0.144029	+	0.989573i	$0.546006\pi$
$374$	0	0
$375$	−1060.39	−0.146021
$376$	0	0
$377$	−329.739	−0.0450462
$378$	0	0
$379$	−1471.60	−0.199448	−0.0997242	−	0.995015i	$-0.531796\pi$
$379$	−1471.60	−0.199448	−0.0997242	+	0.995015i	$0.531796\pi$
$380$	0	0
$381$	−13016.5	−1.75027
$382$	0	0
$383$	2887.39	0.385219	0.192609	−	0.981276i	$-0.438305\pi$
$383$	2887.39	0.385219	0.192609	+	0.981276i	$0.438305\pi$
$384$	0	0
$385$	1754.32	0.232229
$386$	0	0
$387$	−15763.2	−2.07052
$388$	0	0
$389$	8032.30	1.04692	0.523462	−	0.852049i	$-0.324640\pi$
$389$	8032.30	1.04692	0.523462	+	0.852049i	$0.324640\pi$
$390$	0	0
$391$	7498.84	0.969905
$392$	0	0
$393$	−18799.0	−2.41293
$394$	0	0
$395$	78.6488	0.0100184
$396$	0	0
$397$	−2958.09	−0.373961	−0.186980	−	0.982364i	$-0.559870\pi$
$397$	−2958.09	−0.373961	−0.186980	+	0.982364i	$0.559870\pi$
$398$	0	0
$399$	8439.81	1.05895
$400$	0	0
$401$	11859.8	1.47694	0.738470	−	0.674287i	$-0.235549\pi$
$401$	11859.8	1.47694	0.738470	+	0.674287i	$0.235549\pi$
$402$	0	0
$403$	−277.196	−0.0342633
$404$	0	0
$405$	−393.254	−0.0482492
$406$	0	0
$407$	3089.84	0.376309
$408$	0	0
$409$	9048.69	1.09396	0.546979	−	0.837146i	$-0.315778\pi$
$409$	9048.69	1.09396	0.546979	+	0.837146i	$0.315778\pi$
$410$	0	0
$411$	−15197.4	−1.82393
$412$	0	0
$413$	−4309.69	−0.513477
$414$	0	0
$415$	−5259.04	−0.622063
$416$	0	0
$417$	13552.2	1.59150
$418$	0	0
$419$	−12491.8	−1.45648	−0.728239	−	0.685323i	$-0.759661\pi$
$419$	−12491.8	−1.45648	−0.728239	+	0.685323i	$0.759661\pi$
$420$	0	0
$421$	2674.89	0.309658	0.154829	−	0.987941i	$-0.450517\pi$
$421$	2674.89	0.309658	0.154829	+	0.987941i	$0.450517\pi$
$422$	0	0
$423$	−9499.05	−1.09187
$424$	0	0
$425$	−1094.33	−0.124901
$426$	0	0
$427$	1201.69	0.136192
$428$	0	0
$429$	4785.82	0.538605
$430$	0	0
$431$	14304.3	1.59864	0.799321	−	0.600905i	$-0.205193\pi$
$431$	14304.3	1.59864	0.799321	+	0.600905i	$0.205193\pi$
$432$	0	0
$433$	−13331.8	−1.47965	−0.739824	−	0.672800i	$-0.765091\pi$
$433$	−13331.8	−1.47965	−0.739824	+	0.672800i	$0.765091\pi$
$434$	0	0
$435$	1230.05	0.135578
$436$	0	0
$437$	24102.2	2.63836
$438$	0	0
$439$	2642.00	0.287234	0.143617	−	0.989633i	$-0.454127\pi$
$439$	2642.00	0.287234	0.143617	+	0.989633i	$0.454127\pi$
$440$	0	0
$441$	−13173.8	−1.42251
$442$	0	0
$443$	13499.5	1.44781	0.723906	−	0.689899i	$-0.242345\pi$
$443$	13499.5	1.44781	0.723906	+	0.689899i	$0.242345\pi$
$444$	0	0
$445$	−3033.62	−0.323163
$446$	0	0
$447$	−25691.5	−2.71849
$448$	0	0
$449$	−7584.76	−0.797209	−0.398605	−	0.917123i	$-0.630505\pi$
$449$	−7584.76	−0.797209	−0.398605	+	0.917123i	$0.630505\pi$
$450$	0	0
$451$	−7993.65	−0.834603
$452$	0	0
$453$	8520.23	0.883698
$454$	0	0
$455$	402.022	0.0414221
$456$	0	0
$457$	−2116.33	−0.216625	−0.108313	−	0.994117i	$-0.534545\pi$
$457$	−2116.33	−0.216625	−0.108313	+	0.994117i	$0.534545\pi$
$458$	0	0
$459$	−6670.13	−0.678289
$460$	0	0
$461$	10534.8	1.06432	0.532162	−	0.846643i	$-0.321380\pi$
$461$	10534.8	1.06432	0.532162	+	0.846643i	$0.321380\pi$
$462$	0	0
$463$	11067.4	1.11089	0.555447	−	0.831552i	$-0.312547\pi$
$463$	11067.4	1.11089	0.555447	+	0.831552i	$0.312547\pi$
$464$	0	0
$465$	1034.04	0.103124
$466$	0	0
$467$	3678.32	0.364480	0.182240	−	0.983254i	$-0.441665\pi$
$467$	3678.32	0.364480	0.182240	+	0.983254i	$0.441665\pi$
$468$	0	0
$469$	−2278.11	−0.224293
$470$	0	0
$471$	−13488.6	−1.31958
$472$	0	0
$473$	−17395.0	−1.69095
$474$	0	0
$475$	−3517.32	−0.339759
$476$	0	0
$477$	−26130.8	−2.50827
$478$	0	0
$479$	−4779.57	−0.455917	−0.227959	−	0.973671i	$-0.573205\pi$
$479$	−4779.57	−0.455917	−0.227959	+	0.973671i	$0.573205\pi$
$480$	0	0
$481$	708.073	0.0671213
$482$	0	0
$483$	10276.5	0.968112
$484$	0	0
$485$	−7533.09	−0.705278
$486$	0	0
$487$	12663.6	1.17833	0.589163	−	0.808015i	$-0.299458\pi$
$487$	12663.6	1.17833	0.589163	+	0.808015i	$0.299458\pi$
$488$	0	0
$489$	5688.34	0.526045
$490$	0	0
$491$	7219.24	0.663543	0.331772	−	0.943360i	$-0.392354\pi$
$491$	7219.24	0.663543	0.331772	+	0.943360i	$0.392354\pi$
$492$	0	0
$493$	1269.43	0.115968
$494$	0	0
$495$	−11154.6	−1.01285
$496$	0	0
$497$	−2963.41	−0.267459
$498$	0	0
$499$	−5718.31	−0.512999	−0.256500	−	0.966544i	$-0.582569\pi$
$499$	−5718.31	−0.512999	−0.256500	+	0.966544i	$0.582569\pi$
$500$	0	0
$501$	3373.27	0.300812
$502$	0	0
$503$	17206.0	1.52520	0.762602	−	0.646868i	$-0.223921\pi$
$503$	17206.0	1.52520	0.762602	+	0.646868i	$0.223921\pi$
$504$	0	0
$505$	3943.74	0.347513
$506$	0	0
$507$	−17540.6	−1.53650
$508$	0	0
$509$	−18691.0	−1.62763	−0.813817	−	0.581121i	$-0.802614\pi$
$509$	−18691.0	−1.62763	−0.813817	+	0.581121i	$0.802614\pi$
$510$	0	0
$511$	979.400	0.0847869
$512$	0	0
$513$	−21438.6	−1.84510
$514$	0	0
$515$	−1710.16	−0.146327
$516$	0	0
$517$	−10482.3	−0.891708
$518$	0	0
$519$	22682.2	1.91838
$520$	0	0
$521$	−7074.55	−0.594898	−0.297449	−	0.954738i	$-0.596136\pi$
$521$	−7074.55	−0.594898	−0.297449	+	0.954738i	$0.596136\pi$
$522$	0	0
$523$	12647.5	1.05743	0.528716	−	0.848799i	$-0.322674\pi$
$523$	12647.5	1.05743	0.528716	+	0.848799i	$0.322674\pi$
$524$	0	0
$525$	−1499.69	−0.124670
$526$	0	0
$527$	1067.14	0.0882079
$528$	0	0
$529$	17180.5	1.41205
$530$	0	0
$531$	27402.5	2.23949
$532$	0	0
$533$	−1831.84	−0.148866
$534$	0	0
$535$	145.257	0.0117384
$536$	0	0
$537$	−3182.21	−0.255721
$538$	0	0
$539$	−14537.5	−1.16174
$540$	0	0
$541$	2430.45	0.193148	0.0965742	−	0.995326i	$-0.469211\pi$
$541$	2430.45	0.193148	0.0965742	+	0.995326i	$0.469211\pi$
$542$	0	0
$543$	−25130.6	−1.98611
$544$	0	0
$545$	6473.51	0.508797
$546$	0	0
$547$	13829.9	1.08103	0.540515	−	0.841334i	$-0.318229\pi$
$547$	13829.9	1.08103	0.540515	+	0.841334i	$0.318229\pi$
$548$	0	0
$549$	−7640.79	−0.593991
$550$	0	0
$551$	4080.09	0.315459
$552$	0	0
$553$	111.232	0.00855346
$554$	0	0
$555$	−2641.37	−0.202018
$556$	0	0
$557$	−19656.7	−1.49530	−0.747649	−	0.664094i	$-0.768817\pi$
$557$	−19656.7	−1.49530	−0.747649	+	0.664094i	$0.768817\pi$
$558$	0	0
$559$	−3986.26	−0.301611
$560$	0	0
$561$	−18424.4	−1.38659
$562$	0	0
$563$	18649.4	1.39605	0.698026	−	0.716073i	$-0.254062\pi$
$563$	18649.4	1.39605	0.698026	+	0.716073i	$0.254062\pi$
$564$	0	0
$565$	−5364.76	−0.399464
$566$	0	0
$567$	−556.174	−0.0411942
$568$	0	0
$569$	−13392.7	−0.986734	−0.493367	−	0.869821i	$-0.664234\pi$
$569$	−13392.7	−0.986734	−0.493367	+	0.869821i	$0.664234\pi$
$570$	0	0
$571$	−7573.58	−0.555069	−0.277535	−	0.960716i	$-0.589517\pi$
$571$	−7573.58	−0.555069	−0.277535	+	0.960716i	$0.589517\pi$
$572$	0	0
$573$	−26964.8	−1.96591
$574$	0	0
$575$	−4282.77	−0.310616
$576$	0	0
$577$	−12135.1	−0.875549	−0.437775	−	0.899085i	$-0.644233\pi$
$577$	−12135.1	−0.875549	−0.437775	+	0.899085i	$0.644233\pi$
$578$	0	0
$579$	2304.16	0.165384
$580$	0	0
$581$	−7437.79	−0.531104
$582$	0	0
$583$	−28835.7	−2.04846
$584$	0	0
$585$	−2556.20	−0.180659
$586$	0	0
$587$	−11035.3	−0.775940	−0.387970	−	0.921672i	$-0.626824\pi$
$587$	−11035.3	−0.775940	−0.387970	+	0.921672i	$0.626824\pi$
$588$	0	0
$589$	3429.94	0.239946
$590$	0	0
$591$	24910.6	1.73382
$592$	0	0
$593$	2023.70	0.140140	0.0700702	−	0.997542i	$-0.477678\pi$
$593$	2023.70	0.140140	0.0700702	+	0.997542i	$0.477678\pi$
$594$	0	0
$595$	−1547.70	−0.106638
$596$	0	0
$597$	−34031.2	−2.33300
$598$	0	0
$599$	14464.5	0.986651	0.493326	−	0.869845i	$-0.335781\pi$
$599$	14464.5	0.986651	0.493326	+	0.869845i	$0.335781\pi$
$600$	0	0
$601$	3879.55	0.263312	0.131656	−	0.991295i	$-0.457971\pi$
$601$	3879.55	0.263312	0.131656	+	0.991295i	$0.457971\pi$
$602$	0	0
$603$	14485.0	0.978235
$604$	0	0
$605$	−5654.22	−0.379962
$606$	0	0
$607$	−13794.8	−0.922430	−0.461215	−	0.887289i	$-0.652586\pi$
$607$	−13794.8	−0.922430	−0.461215	+	0.887289i	$0.652586\pi$
$608$	0	0
$609$	1739.64	0.115753
$610$	0	0
$611$	−2402.15	−0.159052
$612$	0	0
$613$	16546.1	1.09020	0.545100	−	0.838371i	$-0.316492\pi$
$613$	16546.1	1.09020	0.545100	+	0.838371i	$0.316492\pi$
$614$	0	0
$615$	6833.42	0.448049
$616$	0	0
$617$	−5014.18	−0.327169	−0.163585	−	0.986529i	$-0.552306\pi$
$617$	−5014.18	−0.327169	−0.163585	+	0.986529i	$0.552306\pi$
$618$	0	0
$619$	20464.2	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$619$	20464.2	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$620$	0	0
$621$	−26104.2	−1.68684
$622$	0	0
$623$	−4290.41	−0.275910
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−59218.3	−3.77185
$628$	0	0
$629$	−2725.93	−0.172798
$630$	0	0
$631$	28218.5	1.78029	0.890144	−	0.455678i	$-0.150603\pi$
$631$	28218.5	1.78029	0.890144	+	0.455678i	$0.150603\pi$
$632$	0	0
$633$	45145.3	2.83470
$634$	0	0
$635$	7672.01	0.479456
$636$	0	0
$637$	−3331.44	−0.207216
$638$	0	0
$639$	18842.4	1.16650
$640$	0	0
$641$	−8127.40	−0.500800	−0.250400	−	0.968142i	$-0.580562\pi$
$641$	−8127.40	−0.500800	−0.250400	+	0.968142i	$0.580562\pi$
$642$	0	0
$643$	−11416.6	−0.700195	−0.350097	−	0.936713i	$-0.613852\pi$
$643$	−11416.6	−0.700195	−0.350097	+	0.936713i	$0.613852\pi$
$644$	0	0
$645$	14870.2	0.907773
$646$	0	0
$647$	−21087.5	−1.28135	−0.640675	−	0.767812i	$-0.721345\pi$
$647$	−21087.5	−1.28135	−0.640675	+	0.767812i	$0.721345\pi$
$648$	0	0
$649$	30239.1	1.82895
$650$	0	0
$651$	1462.43	0.0880449
$652$	0	0
$653$	25908.5	1.55265	0.776324	−	0.630334i	$-0.217082\pi$
$653$	25908.5	1.55265	0.776324	+	0.630334i	$0.217082\pi$
$654$	0	0
$655$	11080.3	0.660980
$656$	0	0
$657$	−6227.38	−0.369792
$658$	0	0
$659$	9718.64	0.574483	0.287242	−	0.957858i	$-0.407262\pi$
$659$	9718.64	0.574483	0.287242	+	0.957858i	$0.407262\pi$
$660$	0	0
$661$	20789.9	1.22335	0.611675	−	0.791109i	$-0.290496\pi$
$661$	20789.9	1.22335	0.611675	+	0.791109i	$0.290496\pi$
$662$	0	0
$663$	−4222.16	−0.247323
$664$	0	0
$665$	−4974.50	−0.290079
$666$	0	0
$667$	4968.02	0.288399
$668$	0	0
$669$	−35053.1	−2.02576
$670$	0	0
$671$	−8431.72	−0.485102
$672$	0	0
$673$	31415.5	1.79938	0.899688	−	0.436533i	$-0.143794\pi$
$673$	31415.5	1.79938	0.899688	+	0.436533i	$0.143794\pi$
$674$	0	0
$675$	3809.47	0.217225
$676$	0	0
$677$	18356.1	1.04207	0.521037	−	0.853534i	$-0.325546\pi$
$677$	18356.1	1.04207	0.521037	+	0.853534i	$0.325546\pi$
$678$	0	0
$679$	−10653.9	−0.602152
$680$	0	0
$681$	52260.7	2.94073
$682$	0	0
$683$	−4497.91	−0.251988	−0.125994	−	0.992031i	$-0.540212\pi$
$683$	−4497.91	−0.251988	−0.125994	+	0.992031i	$0.540212\pi$
$684$	0	0
$685$	8957.48	0.499632
$686$	0	0
$687$	−48377.2	−2.68662
$688$	0	0
$689$	−6608.03	−0.365379
$690$	0	0
$691$	−32246.3	−1.77526	−0.887632	−	0.460553i	$-0.847651\pi$
$691$	−32246.3	−1.77526	−0.887632	+	0.460553i	$0.847651\pi$
$692$	0	0
$693$	−15775.8	−0.864749
$694$	0	0
$695$	−7987.80	−0.435963
$696$	0	0
$697$	7052.18	0.383243
$698$	0	0
$699$	−21494.5	−1.16309
$700$	0	0
$701$	12062.9	0.649944	0.324972	−	0.945724i	$-0.394645\pi$
$701$	12062.9	0.649944	0.324972	+	0.945724i	$0.394645\pi$
$702$	0	0
$703$	−8761.48	−0.470051
$704$	0	0
$705$	8960.90	0.478705
$706$	0	0
$707$	5577.58	0.296699
$708$	0	0
$709$	−29560.0	−1.56579	−0.782897	−	0.622151i	$-0.786259\pi$
$709$	−29560.0	−1.56579	−0.782897	+	0.622151i	$0.786259\pi$
$710$	0	0
$711$	−707.252	−0.0373052
$712$	0	0
$713$	4176.38	0.219364
$714$	0	0
$715$	−2820.80	−0.147541
$716$	0	0
$717$	−2880.80	−0.150050
$718$	0	0
$719$	10289.1	0.533683	0.266841	−	0.963740i	$-0.414020\pi$
$719$	10289.1	0.533683	0.266841	+	0.963740i	$0.414020\pi$
$720$	0	0
$721$	−2418.65	−0.124931
$722$	0	0
$723$	21127.8	1.08679
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−11916.5	−0.607923	−0.303961	−	0.952684i	$-0.598309\pi$
$727$	−11916.5	−0.607923	−0.303961	+	0.952684i	$0.598309\pi$
$728$	0	0
$729$	−31365.0	−1.59351
$730$	0	0
$731$	15346.2	0.776472
$732$	0	0
$733$	−13690.9	−0.689883	−0.344941	−	0.938624i	$-0.612101\pi$
$733$	−13690.9	−0.689883	−0.344941	+	0.938624i	$0.612101\pi$
$734$	0	0
$735$	12427.5	0.623667
$736$	0	0
$737$	15984.4	0.798907
$738$	0	0
$739$	−2562.38	−0.127549	−0.0637744	−	0.997964i	$-0.520314\pi$
$739$	−2562.38	−0.127549	−0.0637744	+	0.997964i	$0.520314\pi$
$740$	0	0
$741$	−13570.6	−0.672776
$742$	0	0
$743$	−28880.9	−1.42603	−0.713014	−	0.701150i	$-0.752670\pi$
$743$	−28880.9	−1.42603	−0.713014	+	0.701150i	$0.752670\pi$
$744$	0	0
$745$	15142.8	0.744683
$746$	0	0
$747$	47292.1	2.31637
$748$	0	0
$749$	205.436	0.0100220
$750$	0	0
$751$	38418.9	1.86675	0.933373	−	0.358909i	$-0.116851\pi$
$751$	38418.9	1.86675	0.933373	+	0.358909i	$0.116851\pi$
$752$	0	0
$753$	−16414.2	−0.794378
$754$	0	0
$755$	−5021.89	−0.242073
$756$	0	0
$757$	3302.67	0.158570	0.0792850	−	0.996852i	$-0.474736\pi$
$757$	3302.67	0.158570	0.0792850	+	0.996852i	$0.474736\pi$
$758$	0	0
$759$	−72105.6	−3.44831
$760$	0	0
$761$	8833.40	0.420776	0.210388	−	0.977618i	$-0.432527\pi$
$761$	8833.40	0.420776	0.210388	+	0.977618i	$0.432527\pi$
$762$	0	0
$763$	9155.39	0.434400
$764$	0	0
$765$	9840.82	0.465092
$766$	0	0
$767$	6929.64	0.326225
$768$	0	0
$769$	−6910.59	−0.324060	−0.162030	−	0.986786i	$-0.551804\pi$
$769$	−6910.59	−0.324060	−0.162030	+	0.986786i	$0.551804\pi$
$770$	0	0
$771$	−6289.49	−0.293788
$772$	0	0
$773$	−25699.3	−1.19578	−0.597892	−	0.801577i	$-0.703995\pi$
$773$	−25699.3	−1.19578	−0.597892	+	0.801577i	$0.703995\pi$
$774$	0	0
$775$	−609.473	−0.0282489
$776$	0	0
$777$	−3735.66	−0.172479
$778$	0	0
$779$	22666.6	1.04251
$780$	0	0
$781$	20792.9	0.952662
$782$	0	0
$783$	−4418.99	−0.201688
$784$	0	0
$785$	7950.27	0.361474
$786$	0	0
$787$	−21967.0	−0.994966	−0.497483	−	0.867474i	$-0.665742\pi$
$787$	−21967.0	−0.994966	−0.497483	+	0.867474i	$0.665742\pi$
$788$	0	0
$789$	39800.7	1.79587
$790$	0	0
$791$	−7587.31	−0.341054
$792$	0	0
$793$	−1932.23	−0.0865264
$794$	0	0
$795$	24650.4	1.09970
$796$	0	0
$797$	−33264.7	−1.47842	−0.739208	−	0.673477i	$-0.764800\pi$
$797$	−33264.7	−1.47842	−0.739208	+	0.673477i	$0.764800\pi$
$798$	0	0
$799$	9247.77	0.409465
$800$	0	0
$801$	27280.0	1.20336
$802$	0	0
$803$	−6872.00	−0.302002
$804$	0	0
$805$	−6057.07	−0.265197
$806$	0	0
$807$	−54709.5	−2.38645
$808$	0	0
$809$	−23154.0	−1.00624	−0.503121	−	0.864216i	$-0.667815\pi$
$809$	−23154.0	−1.00624	−0.503121	+	0.864216i	$0.667815\pi$
$810$	0	0
$811$	−27819.4	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$811$	−27819.4	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$812$	0	0
$813$	46543.7	2.00782
$814$	0	0
$815$	−3352.76	−0.144101
$816$	0	0
$817$	49324.7	2.11218
$818$	0	0
$819$	−3615.20	−0.154243
$820$	0	0
$821$	−11310.9	−0.480818	−0.240409	−	0.970672i	$-0.577282\pi$
$821$	−11310.9	−0.480818	−0.240409	+	0.970672i	$0.577282\pi$
$822$	0	0
$823$	−12873.4	−0.545249	−0.272625	−	0.962121i	$-0.587892\pi$
$823$	−12873.4	−0.545249	−0.272625	+	0.962121i	$0.587892\pi$
$824$	0	0
$825$	10522.6	0.444062
$826$	0	0
$827$	12564.4	0.528305	0.264153	−	0.964481i	$-0.414908\pi$
$827$	12564.4	0.528305	0.264153	+	0.964481i	$0.414908\pi$
$828$	0	0
$829$	20023.0	0.838877	0.419439	−	0.907784i	$-0.362227\pi$
$829$	20023.0	0.838877	0.419439	+	0.907784i	$0.362227\pi$
$830$	0	0
$831$	−37678.0	−1.57285
$832$	0	0
$833$	12825.3	0.533460
$834$	0	0
$835$	−1988.24	−0.0824021
$836$	0	0
$837$	−3714.83	−0.153409
$838$	0	0
$839$	−38199.5	−1.57186	−0.785932	−	0.618313i	$-0.787816\pi$
$839$	−38199.5	−1.57186	−0.785932	+	0.618313i	$0.787816\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	4176.30	0.170628
$844$	0	0
$845$	10338.6	0.420897
$846$	0	0
$847$	−7996.69	−0.324403
$848$	0	0
$849$	−41148.5	−1.66338
$850$	0	0
$851$	−10668.2	−0.429731
$852$	0	0
$853$	7748.13	0.311009	0.155505	−	0.987835i	$-0.450300\pi$
$853$	7748.13	0.311009	0.155505	+	0.987835i	$0.450300\pi$
$854$	0	0
$855$	31629.6	1.26516
$856$	0	0
$857$	−41669.4	−1.66091	−0.830455	−	0.557085i	$-0.811920\pi$
$857$	−41669.4	−1.66091	−0.830455	+	0.557085i	$0.811920\pi$
$858$	0	0
$859$	31271.5	1.24211	0.621053	−	0.783769i	$-0.286705\pi$
$859$	31271.5	1.24211	0.621053	+	0.783769i	$0.286705\pi$
$860$	0	0
$861$	9664.42	0.382535
$862$	0	0
$863$	22150.9	0.873727	0.436863	−	0.899528i	$-0.356089\pi$
$863$	22150.9	0.873727	0.436863	+	0.899528i	$0.356089\pi$
$864$	0	0
$865$	−13369.1	−0.525507
$866$	0	0
$867$	−25423.0	−0.995858
$868$	0	0
$869$	−780.463	−0.0304665
$870$	0	0
$871$	3663.02	0.142499
$872$	0	0
$873$	67741.6	2.62624
$874$	0	0
$875$	883.929	0.0341512
$876$	0	0
$877$	1057.91	0.0407332	0.0203666	−	0.999793i	$-0.493517\pi$
$877$	1057.91	0.0407332	0.0203666	+	0.999793i	$0.493517\pi$
$878$	0	0
$879$	55655.8	2.13563
$880$	0	0
$881$	43948.4	1.68066	0.840328	−	0.542078i	$-0.182362\pi$
$881$	43948.4	1.68066	0.840328	+	0.542078i	$0.182362\pi$
$882$	0	0
$883$	1203.63	0.0458724	0.0229362	−	0.999737i	$-0.492699\pi$
$883$	1203.63	0.0458724	0.0229362	+	0.999737i	$0.492699\pi$
$884$	0	0
$885$	−25850.1	−0.981854
$886$	0	0
$887$	25700.6	0.972875	0.486438	−	0.873715i	$-0.338296\pi$
$887$	25700.6	0.972875	0.486438	+	0.873715i	$0.338296\pi$
$888$	0	0
$889$	10850.4	0.409349
$890$	0	0
$891$	3902.41	0.146729
$892$	0	0
$893$	29723.5	1.11384
$894$	0	0
$895$	1875.62	0.0700503
$896$	0	0
$897$	−16523.8	−0.615067
$898$	0	0
$899$	706.988	0.0262285
$900$	0	0
$901$	25439.5	0.940636
$902$	0	0
$903$	21030.7	0.775037
$904$	0	0
$905$	14812.2	0.544058
$906$	0	0
$907$	26715.7	0.978037	0.489018	−	0.872273i	$-0.337355\pi$
$907$	26715.7	0.978037	0.489018	+	0.872273i	$0.337355\pi$
$908$	0	0
$909$	−35464.2	−1.29403
$910$	0	0
$911$	−41003.4	−1.49122	−0.745611	−	0.666381i	$-0.767842\pi$
$911$	−41003.4	−1.49122	−0.745611	+	0.666381i	$0.767842\pi$
$912$	0	0
$913$	52187.5	1.89174
$914$	0	0
$915$	7207.92	0.260422
$916$	0	0
$917$	15670.7	0.564331
$918$	0	0
$919$	−18318.5	−0.657533	−0.328766	−	0.944411i	$-0.606633\pi$
$919$	−18318.5	−0.657533	−0.328766	+	0.944411i	$0.606633\pi$
$920$	0	0
$921$	−62000.9	−2.21824
$922$	0	0
$923$	4764.94	0.169924
$924$	0	0
$925$	1556.85	0.0553393
$926$	0	0
$927$	15378.7	0.544877
$928$	0	0
$929$	−45848.5	−1.61920	−0.809602	−	0.586980i	$-0.800317\pi$
$929$	−45848.5	−1.61920	−0.809602	+	0.586980i	$0.800317\pi$
$930$	0	0
$931$	41222.2	1.45113
$932$	0	0
$933$	6459.22	0.226651
$934$	0	0
$935$	10859.5	0.379832
$936$	0	0
$937$	5905.46	0.205894	0.102947	−	0.994687i	$-0.467173\pi$
$937$	5905.46	0.205894	0.102947	+	0.994687i	$0.467173\pi$
$938$	0	0
$939$	44439.3	1.54443
$940$	0	0
$941$	2680.04	0.0928446	0.0464223	−	0.998922i	$-0.485218\pi$
$941$	2680.04	0.0928446	0.0464223	+	0.998922i	$0.485218\pi$
$942$	0	0
$943$	27599.4	0.953086
$944$	0	0
$945$	5387.69	0.185462
$946$	0	0
$947$	−5804.87	−0.199190	−0.0995950	−	0.995028i	$-0.531755\pi$
$947$	−5804.87	−0.199190	−0.0995950	+	0.995028i	$0.531755\pi$
$948$	0	0
$949$	−1574.80	−0.0538674
$950$	0	0
$951$	−68324.1	−2.32972
$952$	0	0
$953$	−43474.4	−1.47773	−0.738864	−	0.673854i	$-0.764638\pi$
$953$	−43474.4	−1.47773	−0.738864	+	0.673854i	$0.764638\pi$
$954$	0	0
$955$	15893.3	0.538527
$956$	0	0
$957$	−12206.2	−0.412301
$958$	0	0
$959$	12668.5	0.426575
$960$	0	0
$961$	−29196.7	−0.980050
$962$	0	0
$963$	−1306.23	−0.0437100
$964$	0	0
$965$	−1358.09	−0.0453041
$966$	0	0
$967$	36054.7	1.19901	0.599504	−	0.800371i	$-0.295364\pi$
$967$	36054.7	1.19901	0.599504	+	0.800371i	$0.295364\pi$
$968$	0	0
$969$	52243.8	1.73200
$970$	0	0
$971$	−16675.6	−0.551130	−0.275565	−	0.961282i	$-0.588865\pi$
$971$	−16675.6	−0.551130	−0.275565	+	0.961282i	$0.588865\pi$
$972$	0	0
$973$	−11297.0	−0.372216
$974$	0	0
$975$	2411.38	0.0792062
$976$	0	0
$977$	−24732.7	−0.809896	−0.404948	−	0.914340i	$-0.632710\pi$
$977$	−24732.7	−0.809896	−0.404948	+	0.914340i	$0.632710\pi$
$978$	0	0
$979$	30103.9	0.982761
$980$	0	0
$981$	−58213.3	−1.89460
$982$	0	0
$983$	−23711.9	−0.769370	−0.384685	−	0.923048i	$-0.625690\pi$
$983$	−23711.9	−0.769370	−0.384685	+	0.923048i	$0.625690\pi$
$984$	0	0
$985$	−14682.5	−0.474948
$986$	0	0
$987$	12673.3	0.408708
$988$	0	0
$989$	60059.0	1.93101
$990$	0	0
$991$	−29289.2	−0.938852	−0.469426	−	0.882972i	$-0.655539\pi$
$991$	−29289.2	−0.938852	−0.469426	+	0.882972i	$0.655539\pi$
$992$	0	0
$993$	982.107	0.0313859
$994$	0	0
$995$	20058.3	0.639085
$996$	0	0
$997$	28945.4	0.919470	0.459735	−	0.888056i	$-0.347944\pi$
$997$	28945.4	0.919470	0.459735	+	0.888056i	$0.347944\pi$
$998$	0	0
$999$	9489.23	0.300527

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.4.a.a.1.1	✓	6	4.3	odd	2
2320.4.a.o.1.6		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)

\(f(q)\)	\(=\)	\(q+8.48308 q^{3} -5.00000 q^{5} -7.07143 q^{7} +44.9627 q^{9} +49.6170 q^{11} +11.3703 q^{13} -42.4154 q^{15} -43.7733 q^{17} -140.693 q^{19} -59.9876 q^{21} -171.311 q^{23} +25.0000 q^{25} +152.379 q^{27} -29.0000 q^{29} -24.3789 q^{31} +420.905 q^{33} +35.3572 q^{35} +62.2739 q^{37} +96.4552 q^{39} -161.107 q^{41} -350.585 q^{43} -224.813 q^{45} -211.265 q^{47} -292.995 q^{49} -371.332 q^{51} -581.165 q^{53} -248.085 q^{55} -1193.51 q^{57} +609.450 q^{59} -169.936 q^{61} -317.951 q^{63} -56.8515 q^{65} +322.156 q^{67} -1453.25 q^{69} +419.068 q^{71} -138.501 q^{73} +212.077 q^{75} -350.863 q^{77} -15.7298 q^{79} +78.6508 q^{81} +1051.81 q^{83} +218.866 q^{85} -246.009 q^{87} +606.725 q^{89} -80.4044 q^{91} -206.808 q^{93} +703.464 q^{95} +1506.62 q^{97} +2230.91 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 30 q^{5} - 9 q^{7} + 45 q^{9} + 10 q^{11} + 13 q^{13} - 25 q^{15} - 23 q^{17} - 10 q^{19} - 52 q^{21} - 75 q^{23} + 150 q^{25} + 116 q^{27} - 174 q^{29} + 127 q^{31} - 72 q^{33} + 45 q^{35}+ \cdots + 2976 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more